Oh, hi beautiful. You look ravishing today. Okay, so YouTube is just filled, filled with routine videos. Okay, we got morning routines, we got bedtime routines, we got nail routines, fake tanning routines, back scratching routines. I don't know if that's a thing, but it should be. There's a lot of routines, and there's also hair care routines. There are tons of them on YouTube. So we're gonna watch two people's totally different hair care routines, specifically hair styling routines, okay? And I'm gonna have a scoreboard somewhere on the screen, and every time I disagree with something that the person I'm watching does with their hair, I will explain why I disagree, why I think they should have done it differently, and then they will lose a point. That's the game today. We'll see what happens. I don't know, and if you like this, maybe I'll do it again. If you hate it, I will never do it again. I'm gonna share a lot of my knowledge about hair styling with you, and it should be really fun. So let's do it. Woo! Okay, our first video is by Freddie My Love. This is my hair routine care and style, Freddie My Love. Okay, can we get 10 points on the scoreboard? And let's begin watching. So we're gonna skip her hair washing because today I'm just not interested in that. We know a lot about hair washing for my channel. So let's skip to her routine of styling. Wrap my hair up in a towel and pop my pajamas on. Like I said, this is mostly done in the evening, but obviously if it is the morning, then I'll just get dressed in clothes. I just give my hair a rough towel dry, and then I try and leave it for a short while. Ah! I get so worried. No! I'm sorry, Freddie My Love. You just lost a point. Ooh! So soon, we just began the video. I'll show you that again. Ready for it? You're poor hair! So, ah!  You just don't need to do that. Ugh! You're just dangling your hair, roughing it up, really just going at it with that towel. And towels, they're very abrasive. They are so textured. They are really rough. Not good for your hair. Please just scrunch it with the towel. That's all you need to do. Your hair already looks dry. I mean, I don't know what you're really getting out of that with the whole ch-ch-ch-ch. Honestly, the way of getting a lot of moisture out of your hair is letting the towel really soak in the moisture.  This doesn't really do much. Sorry. I mean, there you go. Nine points. And then I'll just give it a final brush through with my Wet Brush Pro because it's just so gentle and lovely at getting tangles out. My hair's always so tangled after I washed it.  But the Wet Brush, great. Love a Wet Brush. Wet Brushes are gonna be just so much better for your hair. They're especially made to work well on wet hair because the bristles in the brush are so flexible. So when they hit a knot in your hair, they're not gonna yank at it. A paddle brush, say, would yank at your hair because those fibers on the brush are very, very stiff and upright. And if you hit a knot with a flat brush, it's gonna really tear at your hair. And when your hair is wet, it's much more fragile, so be careful. So that's great, I love that. One more thing, she is now going in with Elna, Satin. Before I dry my hair, I'm just gonna spray in some of the L'Oreal Element Heat Protectants. Okay, I'm happy she's using a heat protectant. I don't know anything about that specific one, but I've done a lot of different experiments on heat protectants because of my own haircare line. That's a long story. But what I can say is my findings were that when you use them, they work. So that's good. So I'm going to be using my GHD miniature travel size hair dryer, just because it's so lightweight. Then I just give my hair a breath blow dry until I can say it was almost about half dry. Okay, now listen, I'm gonna be really hard on these people in this video just because it's more fun that way. I can't give everybody a really high score, so I'm just gonna casually pick them up hard a little bit. Love you guys, thanks for being in my news. Even though you didn't volunteer to be. GHD blow dryer, cool. Never use that one, seems legit. I do, however, not like that nozzle at all. I would have put a much more concentrated nozzle, but I'm glad she's using a nozzle in the first place. I would have liked if she prepped her hair with more product, just because she has blonde hair. She needs a little bit more love. Just something more to hydrate and smooth would have been great before she blow dried.  So applying these products before the blow dry, I personally believe is better. So because she didn't apply more than a heat-sailing protector, we're gonna take off one point from this chorus. That brings her down to an eight, which is still a good number, okay? And I, again, I'm being very hard, but let's keep watching. There are many different products, but the one kind of standout product that I always find myself going back to and repurchasing is the Phyto Paris. I think that's how you pronounce it, exceptional cream. So interesting. She kind of like blow dries 80% and then puts the product in her ends after. I'm glad she's putting product in. I'm happy about that. However, I do wish she put it in when it was a little bit more damp. And I just put a few pumps of this into my hand and just spread it kind of evenly through the lengths and ends of my hair. Then I will just continue blow drying my hair. And I've switched brushes because you aren't supposed to use the Wet Brush Pro with heat. And also this brush gives your hair a really nice smooth finish when you blow dry with it. And then once my hair is almost dry, I just throw it upside down and blast some air into the roots because this really just gives it a bit more volume. And then once it's fully and perfectly dry, I just give it a final brush through. Oh, and it's just the nicest feeling in the world having clean, beautiful smelling, soft, lovely hair. I just, okay. So here's the thing. This girl has fabulous hair. It's not frizzy. It's quite beautiful.  I have pretty much the same hair as her so I totally understand. Blow dries amazingly with little effort. And if she air dried it, it probably come out, it's very similar to this. But I would prefer if she used sections when she blow dried just to make sure everything is neatly organized. And I feel like the blow drying process is just faster when you section properly. I'm not gonna take a point off for that because that's just not necessary for her hair but it would be just faster for her. I like to do halo section. So just take out this center part to clip it up and do all around the outside first. That way you get all the frizziness from up here and down here out. And then you can move on to this part of your head which is not as curly and not as frizzy as this part. But that's fine. If she likes doing it that way, it's all good. I like that she flipped her head upside down and blew cold air in because it's gonna make her hair style stay a bit more volumous after she's done. But if you want even more volume, definitely use a round brush next time. Oh, it's the best feeling ever. So then to style my hair, I mostly use my L'Oreal steam pod. We worked with basically any tool. But I do find myself going for L'Oreal steam pod because it's very unique. It uses water and steam to create heat. Okay, exactly. It can do this technique with basically any styler. And then once I get to the end, I just curl it up and release. So basically the ends of my hair flick. And then whilst the hair is still hot, I use my fingers to roll the hair up into a curl and pin it to the top. So once I've done this and all of my hair is pinned up and I have this really cool hairstyle, I just use some hairspray. Wow, spray it right on her face. It doesn't go hard or sticky or horrid. The reason I like doing this at night is because I like to put my hair up in a silk turban. Because when you're in bed lying on a pillow, your hair is really prone to friction, which causes frizziness and breakage. And it also just keeps my curls in place. Because if I didn't wear the turban, they would definitely blow up. Let's see what happened. So it just keeps the curls in place. So the next morning, I will just take all of the pins out and let the curls loose. And I just absolutely adore the way this makes my hair look. Everyone thinks I've gone and had it blow dried and I hardly ever do. It just looks really effortless and. Okay, that was a lot. Let me break it down real quick. So she uses the steamer, which I've honestly never used a hair steamer before. Steamer's like quite an interesting contraption. And she kind of takes it and runs it really fast until she gets the ends and then flips her ends. What I think she's looking for is a really volumous blowout kind of look without actually doing the blowout, which is quite confusing to me. Watching this, I feel like she goes through a lot of steps to just get a very simple look. If I were to do her hair and she wanted this look, I would just recommend learning how to do a nice round brush blowout on yourself. It's really not that hard with practice. You can do wide sections at a time. You can then pin it up after you can wear it to sleep and you will get a much more voluminous, much more gorgeous, fabulous blowout. What she ended up with was beautiful. However, it could be a lot better and a lot simpler for her if she just did a simple round brush blowout. The other alternative for that, which is also easy, is doing everything she did, blowdrying it really fast, just getting the moisture out of her hair and then taking hot rollers and putting the rollers in her hair overnight. Now that can be uncomfortable, so I don't blame her if she doesn't wanna do that. So I understand that, but I just think a simple blowout is the best decision for her and it just honestly would have been a better outcome with less work. So for that, I'm gonna take one more point off for you. That's a seven. We are at seven points, still not bad, planning on being hard on people today and I feel like seven's pretty good still. So let's keep watching until the end and let's see what else she does. There's still more, yeah. So for the finishing touches, I'll usually take a straightener and I basically just go over the top bits of my hair in case there's any kinks there from where I had the curls pinned up. And then finally, I like to put some oil or serum into the ends of my hair and I'm using the Body Shop Beautifying Oil. I just throw a couple of drops into my hands and just run it through the ends of my hair.   I was gonna say, girl, you got patience.  Good for you. Overall, the look was beautiful. I'm not saying this girl has bad hair. She's her hair poorly. It's beautiful. She has great hair. It's awesome. I commend her for all the work she puts into this. The straightener at the end did bother me slightly and I know why she's doing it because she had those kinks in her hair from putting the pins in her hair. If you make sure it's 100% dry before putting those pins in and holding those curls, it shouldn't really make a kink in your hair. It also might be because of the pins you're using. Try and use pins that are flat and don't use bobby pins. Those will make it easier for your hair to form into place and not get any ridges and bumps on your roots. The only reason why I don't like that you flat ironed after is because you're kind of depleting a lot of that volume you had just kind of put in with that pin curl set. Nobody really likes flat hair when it's supposed to look like a beautiful volume is blow out, you know what I mean? So for that, let's just take one more point off and leave her at a six. That is Freddie's number, a six. That is not bad. Again, you didn't do a bad job. I'm just nitpicking for the fun of it. Let's move on to the next person. Let's see if they will beat Freddie's score or we'll do worse than Freddie. Okay, and next we have something a little bit different. This is a wavy hair routine. This is by Dolce Candy. My wavy hair wash and go routine. She's styling her hair with a sort of natural vibe. So let's see how she does it. Slay those curls, wavy hair. Okay. And now I've ditched my towels and I've upped it out for a 100% cotton t-shirt. Yes, Dolce! That I have. To dry my hair, I'm just gonna start by scrunching out all of the excess water. Oh, can we get a 10 on the board over here? Boom. So she is using a cotton t-shirt to dry her hair. Oh, do I love to see this? I really do. I love it, I love it, I love it. This is awesome. Cotton t-shirts are just gonna be so much more gentle than an abrasive bath towel. It's just gonna be so good for waves because it's most important with waves to keep the wave fully intact and encourage absolutely no frizz to happen and using a gentle fabric like cotton is going to ensure that you get no frizzing. This is awesome, this is so good, okay. So now that most of the water is out of my hair, I am going to lay the same t-shirt down like so and I'm actually gonna wrap my hair in this t-shirt. It's going to encourage all of the excess water to fall out and the curl is going to form a little bit better. I have no product in my hair yet, but it's also going to help my hair dry quicker without any frizz. Yo, how did she just do that? What, that was the coolest I've ever seen. What, that was like genius the way she just wrapped her hair. Wow, I'm shook. Oh girl, you got me on that one. I've never seen anybody do that before. Can we add another point to the board? No, she already has 10, she's still at 10. This is awesome. Now that my hair is nice and damp, I'm ready to start styling my curls. Now I'm going to be using the Diva Curl frizz-free volumizing volume, cool. This is going to add volume to the roots of my hair, which is something that I learned that I need to apply because previous times my hair started to lay flat because of the product was weighing down the curl. So this is gonna give it a little bit of a lift on the roots and I just apply a few pumps and just massage it onto the scalp. So, she's losing a point. Oh, we're down to nine. All right, because once again, I'm very hard on people today, but just there's a lot of this going on and a lot of shaking that beautifully formed wave you just made with this delicate drying technique you use of wrapping your hair in a T-shirt. You're kind of messing that all up right now.  Just get it all on your roots, but just don't frizz it up. That is the last thing we want, waves is frizzing and that's what's kind of going on right now, which is scaring me. Now I'm taking the can to coconut curling cream. This stuff is amazing, but I have learned that a little bit of this stuff goes a long way. I'm using about that much and then I just rub it through my hands and my fingers and I applied only to the bottom portion of my hair. So also starting from the ears and below and this is going to help my curls stay nice and defined. Okay, so I really liked the way she scrunched her hair there. It was quite beautifully done. The product looked like a very heavy, hydrating formula, sort of like a leaving conditioner probably, which is great for waves because wavy or curly hair is pretty much always drier than straight hair. And now she's going in with another product. Next thing I use is the Diva Curl Ultra Defining Gel. This is another product that I use to eliminate frizz and to scrunch my hair. Once you dry your hair with either a diffuser or you air dry it, it's going to create a cast. So you have to release the curl from the cast by scrunching it once the hair is dry. So I am just going to apply this the same way, scrunch it up just like that. I used a little bit on the front as well. I love that she said that because a lot of people think that when your hair dries kind of hard with a product, it's as if product's not good or it's making your hair too crunchy. The thing is when you're using a holding product for curls, such as my product Glitterati, it can form a slight cast on the hair. Some people like that cast. Some people do not like that cast, which is fine because you have the option of breaking up that wave after. You can still break it up and those curl defining properties from the product will still last in your hair. You can make your hair soft by running your hands through it, breaking the cast up and it will still stay intact because the product has memory. It is laying on top of that hair and helping that wave stay. So that's fine. I love that she said that. So here's a little trick that I learned for volume from the Curly Girl, the handbook. It's a really awesome book all about curly hair. And what you do is you take a section of hair and you lift it up here at the root and then you just simply clip it down. So you can see the hair has some volume and I do this all around the crown and this is going to lift and give me volume when I dry it. This is advanced. This is going to leave her hair with such great volume. You have the opportunity here to really manipulate the way your hair dries and get those really nice waves and volume right at the roots that you wouldn't probably normally get. And I love using my Dyson blow dryer because it's very powerful and it's also very light. For me, it takes forever to dry my hair with a diffuser because I do not use any heat whatsoever to dry my hair. I want to stay away from that. So I'm using only cool, cold air. No, no, don't say. Sorry, I just got like flashbacks. I've had models tell me that I cannot use heat on their hair before and I get so annoyed. They're like, can you blow dry my hair with cool air? And I'm like, no, actually I can't. No, I can't. I mean, cool, if she can do it, more power to her. That just seems like not exactly necessary. You know, it's the intense, intense heat that we just stay away from. Otherwise, I don't think you really need to be using cool air, but if that makes you happy and you have the time just been doing that, you don't want to go for it. I'm not going to take a point out for that because that's just really personal preference on that one. But it does irk me a little bit. Also, heat really helps activate product and form curl. So I just feel like heat would be the better option here. Should I take a point out for that? Should I? Should I? No, I'm not gonna. I'm feeling nice for this very one second of this video. It starts by just sort of going up and down with a diffuser for a few times just to kind of get the hair dry a little bit. And it's also in medium speed because if not, it's going to give me more frizz. So to me, this... That's also great medium speed because if she has it on high, it's gonna create a lot more frizz. I usually have it on like low when I'm doing wavy hair with a diffuser because what will happen is when you're holding the diffuser down here and pointing it up towards your hair, it'll take all those little baby hairs and kind of like make them fly up and it'll create a lot of frizz up here. So I usually, when I'm using a diffuser, I always try to blow dry downwards actually. So I don't exactly love that she's picking up the hair with a diffuser and kind of like going like that. I kind of like to guide the hair into it from the side. So I kind of like take the hair, push it into the diffuser over here and hold it there with my hand. That way you're not creating more frizz this way. We're gonna take a point out for that. I feel like it. You know what? I've been too nice. I don't like that technique, okay? I'm sure it'll come up fine, but I just think there's a more optimal way. So we're taking a point off. Sorry. Okay, Dolce, you lost the point. We're out of an eight now. Okay. I also just add a little bit of the cold air to the roots just to dry them so I can take out those double prong clips that tend to get in the way. So just add some cold air directly onto that. I see a little bit of frizz, but it's nothing serious. And then the trick is just to, and then it can dry. I like the way she's doing this. I love it. I really love it. It evaporates and escapes. So that's basically all I do for my hair. She's such gorgeous hair. Just blow drying my hair some more. So now my hair is about 80% dry, which is the maximum point of drying is that I like to leave my hair because it just gives me beautiful frizz-free curls.  I love that she said she doesn't dry her hair fully. She leaves it 20% wet, 80% dry, and then she leaves it to dry the rest naturally. You're just not gonna get that perfectly shiny frizz-free look unless you just leave it a teeny bit damp when you're done with the diffuser. It's a great tip. I love that. What I'm doing now is just scrunching my hair with my dry hands, no product, to release the curls from the gel cast. So here you have it.  It looks so pretty. Okay, that was dope. Okay, Dolce Candy ends it up with an eight. So Miss Dolce Candy, congratulations. You won today's competition. You have the better hair care routine. And again, this is me completely nitpicking, completely trying to find things wrong with these people's hair care routines. Those two were actually quite great. I didn't see anything totally out of the ordinary, crazy, totally wrong, but really impressed with Dolce on that beautiful wavy hair care routine. It was fabulous. So congratulations, you're the winner. You get absolutely nothing. Woo, that was fun. I actually really enjoyed that. I feel like we all got educated a little bit. I learned some things. I'm sure you guys learned some things. And it was quite fun. So that's all I got for you today. If you guys wanna see more videos like this, please let me know. Also, if you guys want to try out some amazing, fabulous, glorious hair products to bring your hair to the next level, please check out xmonohair.com linked below. Check out xmonohair on Instagram for updates on new product launches. And follow me on Instagram and Twitter at bramanomyc where I just post pretty pictures of myself. And that is all for today. Thank you guys so much for watching.  And I'll see you next time. Bye.


Hi everyone happy Friday and happy new release day happy immaculate day Coming at you with 2020 Panini immaculate football six box pick your team number one from Jasper's case breaks calm No fillers no camera either. Let's get this camera turned on No fillers no number blocks or anything like that. We filled it up the old-fashioned way straight up So thanks very much everyone for giving this shot on Friday the 18th immaculate football some good stuff here Cameron with that last spot mojo dolphins Good luck to him and there's the first case right here. Hopefully it's a good one. Those are the colors of the boxes I kind of like that Got like a berry color here with that sky blue All right, so I'm gonna stack all six you can see on the the top camera right there all six boxes stacked here And here we go folks It's sort of a Christmassy look maybe a Christmas red with a little little snow right there We've got a redemption on the bottom here. All right Put a little blank card on top to hide the hits a little bit. There's our first card 37 out of 60 Phillip Rivers for the Colts Derek Disser 10 out of 10 That's nice John Riggins Hall of Fame patch for the Washington football team that goes to Derwin Player warm, but still some good color in there for Washington Next up we got 14 out of 20 Zack Moss Nice patch. He's got a cool autograph That'll be for Buffalo Robert my hill with the Buffalo Bills Buffalo These are nice.  I think this is a Kind of a clever idea if you think about it. It's a good alternative to stickers, right? And another one of these eye black autographs 20 out of 99 rookie eye black jersey and autograph Jordan Love Silver ink on this one. It goes to Paul Townsend silver ink Silver ink.  That'll be for Alifanzo there it is. That's box one and pick your team one Next one Greg saying your readers. He is tough to watch it is man But helps your finza. Yeah, Teddy Jaspie is a Dolphins fan and he's setting up on Instagram pretty soon But yeah, he's a Dolphins fan and he was just like, yeah, sorry Joe gonna root against your readers today help help the Dolphins playoff push. All right, we got only slide this blockchain card off the bottom here And it's Brian Urlocker immaculate Hall of Fame signatures for the Bears.  I might take Considering taking the investing in the New England Patriots this weekend might have my official picks tomorrow. Andre Johnson 50 out of 60 That'll be for Arthur Hackett and the Texans We've got the standard relic Alan Robinson a little bit of a dent there 10 out of 25 the Chicago Bears That's gonna be for Stephen Wright. We've got a nice patch here speaking of the Patriots 20 out of 26 Sony Michelle here's the thing the Dolphins Dislike stopping the run for some reason They're good they're good at stopping the pass, but guess what Patriots can't pass I'll go to the Patriots. That'll be for Karen and her pass sub T.J We've got my rivals the Broncos premium relic and autograph Clinton Portis eight out of 49 Jeff Gordon With the Broncos. Yes race car driver Jeff Gordon joins her break We've got Michael Pittman, Jr. Rookie shadow box autograph 51 out of 99 I'll go to the Colts Derek disser. He's pretty good our Colts a sneaky playoff Like picked to go deep into the glass And oh nice nice time to hit a Jalen Hertz 55 and a 90 90 he's getting another start That's for Paul Townsend the Philadelphia Eagles Fly Eagles fly. He's got a nice auto too. I don't know if I have an upset pick Tony Often I'll think about that too Anyone have a good upset pig? Darnold to the Colts Mikey is saying I don't know Frank Reich I feel like it's more likely that Frank Reich tries to get I mean if Rivers doesn't play another year But it hard time believe the Frank Reich is not going to chase after maybe Carson Wentz They've got a good relationship David T is saying chiefs lose this weekend. That's his upset upset special Another blockchain card on the bottom and it's Jimmy Garoppolo and Debo Samuel another San Francisco Redemption Upset special chargers wins as Sean Jeff. Who are they playing next week? David says I'm a real football fan from Scotland a real American football fan or football football It's Calvin Johnson 16 out of 60 Sean you got to pay attention buddy. I know it's tough for you to read, but you got to pay attention We're talking about upcoming upsets 76 out of 85 Chad Johnson bangles and that wasn't really much of an upset special was it? It's a coin flip game too bad team playing against each other Sean Maddock and the bangles Hardly an upset. We've got Kyler Murray Nice 22 out of 25 Carl with the cards It's the remarkable Kyler Murray relic and we've got oh Look at that Wow dual eye black autograph Jim Kelly the past and Josh Allen the present and the future maybe Eight out of 15 Buffalo Robert Myhill with the Buffalo Bills that is awesome Ding-ding. I don't know where where's my bell Sean? There you go. All right Next up we got Grant Delpit 22 out of 99 for the dog pound that's gonna be for Daniel Anderson and Cleveland Cleveland this is for you and we got Jerry Judy three color patch an autograph for my rivals the Broncos It's nice patch though 37 out of 75 Denver Jeff with Denver. All right. We're a few boxes in we got a few boxes to go Look at this card right here Piece of a helmet cleat. All right. We got 21 out of 25 Randy Moss. You got Moss That's Patriots edition going to Karen. We've got a triple relic J. K. Dobbins for the purple birds Out of 49 Goes to Nico and the purple birds Chulases maybe Greg saying. Oh nice Nick. I'm glad you got a jazzy shipment safe and sound Speaking of shipments folks remember during the holidays and on with the pandemic piled on top of the holidays a lot of stuff Is being delayed or at the very least Tracking numbers aren't being updated so always just just keep that in mind if you receive a tracking number Pretty much that means it's out the door at Jaspie's and it's on its way It may look like it's stuck in like Hermosa Beach or LA. It's not it's on its way to you So we appreciate your patience on that and be patient with the with the post service to their working their butts off All right. Here it is. The big thick one is Dan Marino sneak peek look at that sure what part of the shoe that is but You can see a little bit of dirt right here. Wow It's kind of light too. Usually some of these are a little bit heavier, but that one feels a little light Cameron Hutton Got the dolphins straight up and gets the out of five Dan Marino sneak peek Maybe the top of the shoe Aaron's thinking Either way you can definitely tell that it's game used Game used think about that Dan the man's foot wasn't there There you go Cameron out of fives and under get the train whistle. So All aboard the big head Express Not Tremaine Edmonds Patch and autograph that kind of looks like it's game used as well 39 out of 99 Buffalo I'll be for Robert my who picked up the Buffalo Bills. Yep game worn Game worn slash game used and then down here. We've got Alejandro Villanueva Nice triple relic and autograph 26 out of 99. That'll be for steel curtain and his Steelers What's down here? What's why is that autograph up? So there it is nice van Jefferson? One out of 25 two-color patch. He's got a nice auto to for the Rams whose house Rams house Kurt disser With the rammelie with van Jefferson, right? Whose house Rams house whose house? Rams house Right two more to go and then we got another case coming up Pick your team to come up right after this and like I promised if pick your team three sells out before I'm done with pick your team, too I'll do it. I'll do it before I have to go through a bunch of orders. All right We got Keenan Allen 53 out of 60 That is for the Chargers. That's gonna be for Jack. We'll find the guy that throws him the ball We've got John Elway Hall of Fame relic to 49 that will be for Denver. That's gonna go to Jeff Wow a one-of-one Tredavious white Nike swoosh Is that a bit of a maybe field paint on there Robert my hill with the bills game warm material Buffalo and all aboard the Big Head Express That's pretty sweet. We got a redemption. It's Wow Jerry Rice Larry Fitzgerald and Randy Moss immaculate Trio's autograph Let's actually check what addition I'm assuming it's Niners edition of Jerry Moss and Jerry Moss Jerry Rice and Vikings edition of Randy Moss, but let me just double-check sometimes Sometimes that they go a little goofy on that Immaculate Trio's There it is card number four. It's San Francisco, Arizona, Minnesota. All right, so we'll do a randomizer at the end One happy person two sad people on that one Jonathan Taylor silver ink autograph 79 out of 99 That'll be for Derek and the Colts and we've got for the purple birds Devon Duvernay 99 78 out of 99 three color patch an autograph good-looking patch in there to for Nico and The Ravens right one more box ago and that that sad sad randomizer whoever gets that. Can you please redeem it? I would appreciate that if you redeem it and then shoot it Send us a picture of it or tweet us a picture of it at Jasper's brakes. We'll retweet that that's got to look pretty cool Good luck, everybody. We got a one out of five base card. It's Keenan Allen Nice Keenan Allen going to Jack and the Chargers Jack All aboard the Big Hit Express We've got a piece of Keeshawn Vaughn's helmet. That's awesome 10 out of 10 This might not fit in a 190 top loader. This goes to Demetrios and the Buccaneers. Yeah, I'm gonna put this into a slider box Put a couple blank cards in here so it doesn't rattle around but that's a good-looking piece of the helmet We've got a Zach bond for the Saints nine out of 99 and that'll be for Ryan Cason and Nolence Actually, Nick's like there's probably less of those Less than 50 of those triple autos. I'm actually looking at cardboard connection.com. There's only five Immaculate Trio's autographs. No for Immaculate Trio autographs, and they're all numbered to five So someone's gonna get a train whistle. Look at that Ryan Fitzpatrick looking good Unforgettable fashion show autograph 23 out of 25 Buccaneers that's gonna go to Demetrios We've got Chase Claypool. He's been having a nice season 86 out of 99 silver ink autograph. That's for steel curtain and his Steelers Love that this I like this set right here. Looks pretty good in the final one. Oh That's nice. Chase.  He pretty much took over that game chase young For the Washington football team Derwin looks like he's gonna be a force in the league for years to come all right now time to make one person happy and And Time to make one person happy and two people sad It's the Jerry Rice Larry Fitzgerald Randy Moss Triple autograph. I don't have any other randomizers to do. All right. Well, that's yeah a good point Sean Jaspy You're right actually he's usually a redemption So that's the rare live auto siding All right now. Let's go back to random org. There's some new dice right there and And there's the blank list. We've got San Francisco, Arizona and Minnesota It's gonna be team on top after Four three and a one four times One two Three and Fourth and final time after four times goes to Arizona after four Carl I don't know if I think I'm maybe just called your name once all break and There you have you're gonna get the Jerry Rice Larry Fitzgerald and Randy Moss Triple autograph my apologies to San Francisco Alifonzo and Minnesota David, but We'll get him and try to get him next time Thanks very much everyone, but congrats to Carl and congrats everyone who hit in this if you didn't We got some more immaculate football on Jaspy's case breaks calm. Let's let's keep chasing. I'm Joe I'll see you next time for the next one. Bye. Bye


which is, do we have the will to stay in contact with the rich humanity of black people? To stay in contact with the beauty of black people, the intelligence of black people, the tenderness of black people, the soulfulness of black people. When you talk about Brother George, Perry Floyd Jr., you're talking about a human being. You're talking about a precious son of George Perry Sr. Here we are in 2020, saying very much how you started this show, my brother. Do we have what it takes to really acknowledge the rich humanity of black folk to treat us fairly, to treat us like human beings? Because if not, you're gonna lose your democracy.  For one for black voters, especially black women voters, you would have gone fascist under the gangster named Trump. Tell you brother, if the police can't do it, then we're gonna do it ourselves. Let me tell you this too. Let me just add just one thing though, brother. Yes, sir, go ahead. I thought about Brother George and I thought about my brother and I thought about my father and I thought about my grandfather. I'm not gonna stand there for no nine minutes and 29 seconds and watch somebody murder my father. No, no, uh-uh. No, no, no. I believe in non-violence, but I'm not gonna watch that kind of murder. I love my brother Charles McMillan. That's why he was crying. That's what his tears were about. He felt helpless. We not gonna do that.  We're gonna have to intervene in some way. They ain't gonna kill us like that and we remain spectators. We got to self-respect. We got to self-defense and we intervene when you start killing us like that. I'm sorry. Welcome back, everyone.  But first, give me 30 seconds to tell you about this new free offer for my viewers. Now, I'm pretty sure we all know that inflation is killing our retirements, dead. The real rate of inflation? Well, it just won't let you keep up or make a profit. You need a different approach. Precious metals, not Robinhood or stocks that can crash and leave you losing money. An IRA or a 401K with Noble Gold is the answer. And this month, Noble Gold is gifting a genuine, rare, Carson City minted Morgan Silver Dollar with every qualifying IRA or 401K. You can find out more by clicking the link in the description or pin comment or by just going to noblegoldenvestments.com. Make sure to tell them Drone Tech politics sent you. Do we have the will to stay in contact with the rich humanity of black people? To stay in contact with the beauty of black people, the intelligence of black people, the tenderness of black people, the soulfulness of black people. When you talk about Brother George, Perry Floyd Jr., you're talking about a human being. You're talking about a precious son of George Perry Sr. I just love that he talks about stereotyping black people, yet he always speaks in these broad terms saying us and them. Both Lemon and Cornell talk about black people as if they're some sort of monolithic hive mind. And they pretend like things that happen to individuals because of their own choices are somehow happening to all black people when the stats and the data are very clear that that's not the case. First off, Cornell broadly judges and condemns America for quote, not seeing the rich humanity of black people, the beauty of black people, the intelligence of black people, the tenderness of black people, the soulfulness of black people. I don't know about you, but that sounds like stereotypes that he's promoting. How could you possibly judge every person that way based on the surface level feature of skin color? And he's saying all these things in an apparent attempt to deflect from who George Floyd actually was. Here we are in 2020 saying very much for how you started this show, my brother. Do we have what it takes to really acknowledge the rich humanity of black folk that treat us fairly, the treat us like human beings? Because if not, you're gonna lose your democracy. Is it just me or did that sound like a threat? Apparently we're supposed to get rid of trials and just acquiesce to mob violence. He talks over and over again about the humanity of brother George Floyd. But what about the humanity of George Floyd's victims? How about the humanity of the pregnant woman that he held at gunpoint as a hostage? Did her humanity matter?  The vast majority of human beings don't invade homes, much less take pregnant women hostage at gunpoint. Nothing that happened to George Floyd was anybody's fault, but George Floyd. Cornell keeps talking about his brother, his grandfather, his father, but why would any of those people ever get themselves into the situation that George Floyd did? They wouldn't.  That's a lie. It's an inflammatory lie that's gonna lead to much violence. The stats are very clear.  And on top of that, there were around 400 white people shot by police. Oh, but that's just normal operating procedure.  Tell you brother, if the police can't do it, then, you know, we're gonna do it ourselves. Okay, let me tell you this too. Let me just add just one thing though, bro. Yes, sir, go ahead. I thought about brother George and I thought about my brother and I thought about my father and I thought about my grandfather. I'm not gonna stand there for no nine minutes and 29 seconds and watch somebody murder my father. No, no, uh-uh. No, no, no, I believe in non-violence but I'm not gonna watch that kind of murder. I love my brother Charles McMillan. That's why he was crying. That's what his tears were about. He felt helpless. We not gonna do that.  We're gonna have to intervene in some way. They ain't gonna kill us like that and we remain spectators. All right, so this is where we're getting into dangerous territory. This rambling, incoherent nut job is now inciting violence against the police or at the very least giving out justifications for attacking the police. How about this? It's a crazy idea. How about instead of justifying and encouraging people to fight the police or spectators to attack police which is clearly insane and unhinged from reality, why not encourage these people to sue the departments if they're mistreated or unlawfully arrested? I've talked about it before but on YouTube there's lots of these First and Second Amendment audits. These people flex their rights to police officers to see how they react. If the police officers mistreat them or run rampant over their rights, these people sue the departments and let me tell you something, they end up successfully suing these departments for tens of thousands of dollars. Seems like that would be a much better outcome to people fighting the police and then dying and doing so. Oh, but I have a feeling that's exactly what people like Cornell and Don Lemon want to happen. We got to self-respect. We got to self-defense and we intervene when you start killing us like that. I'm sorry. Dear Lord, why is he crying? Is that even real crying? I don't see any tears. In fact, what's with all these fake journalists crying all the time now? You know who constantly tries to manipulate me with crying? My kids. If they don't get what they want, they cry in an attempt to manipulate me because they know that sometimes it works. It almost seems like our former free press is adopting the tactics of children, the old logical fallacy of appeal to emotion. They obviously do this because the facts simply aren't on their side. Not only has murder not been proven in this case, there's exactly zero evidence that it had anything to do with race. If you disagree with anything I said in this video, I'm open to have a conversation. You can find my discord link in the description or pin comment. Catch me on there and let's talk about it. All right, that's all I have for this one. If you enjoyed it, please hit that like button, share and subscribe and I'll see you all next video.


How's it going, fellas? We're playing every single NBA 2K. And we gotta get a perfect release on all of these. Now, in the early ones, it's not too hard. You just have to choose Michael Jordan or somebody. But as we go on, I reckon they'll get a little bit more difficult. Up first, we have the original NBA 2K with Alan Iverson on the cover. We're playing on Dreamcast, which means we got these butt-ugly controllers with one analog. So let's get into it. Have you played this game before? Never, why can't I? All right, should be easy W done. You know, after doing the NBA Live video, this isn't so bad. This is not so bad. Oh, actually, hold on, hold on. We gotta get the difficulty on point if we're gonna do this. All starters, the highest difficulty. We're gonna do that on every single game. Highest difficulty simulation. I am an all-star, but... I forgot, I forgot how to play! Hey, I... Let's get it! Let's get it, boy! That's not the hardest difficulty.  It was! All right. Okay, I guess I lost. Ha-ha! All right, we're up next to NBA 2K. Everyone, Alan Averson is also on the cover of this one. Now, I am sweating, but it's because it's hot as s***. I don't know what is going on. It's winter, but we're in Atlanta. You're privileged. I didn't give it away to your mother. Okay, I'm calling it over to Alan. Is that Patrick even? Sure. I did not know he played for the Sonics. I don't know how to jump. I don't know... Oh no! That's game! Ah, let's go! That's game! This is looking like your first... This is the hardest difficulty. All right, man. Okay. I called it with Reggie Miller, too. I know you did. I heard you call it. All right, next up, fellas. I'm on a losing streak, NBA 2K2. Has Alan Averson on the cover as well. This one should be a little bit more challenging. The early ones were super, super easy, but as we go on, the games do get harder. New rule! If you hit a three, kind of like overtime in the NFL, I have to hit one or I lose. Okay, okay, that's fine. You know what I mean? Bad, that's way more fun. Ha-ha-ha!  I do remember this now. Okay, I just need shooters. I need shooters. Yao Ming, let's get it. You're not shooting up for the reason, Yao Ming. Imagine he hits it, you're gonna cry. I would cry if you hit it. That's me, baby. Ah! Steve Nash with the... You can't leave me open! Ah! You gotta answer back, bud. Derek Fisher, headband Derek Fisher? Oh, wait, wait. Is that Rick Fox? I saw him in the game one, in his shoulders. Kobe! Ooh! It's quick! It's quick! I'm still 2-1-1 though, so we're good. Bro, this man was on every cover. What is this? NBA 2K3? Guess who's on the... It's Alan Iverson again. 99,001 and 2-0-3. Guys, the go-way. Who's in the next one? Oh, this is T-Mac. Although it would have been hilarious if it was Iverson. That's AI, bro. That's a T-Mac. Oh! T-Mac, what the hell? Does this look anything like T-Mac? Yes, the answer is yes.  You better pay attention, my guy, because this is Ray Allen and Kobe on the squad. I know. Yeah, I mean, are you shooting at him? Oh, yeah, oh, yeah. Square. No sir! No sir! No! I'm right there. I'm there. Boards! He just tipped it in. Oh, I know, what the hell? That was air ball. Oh!  Oh, T! You know what? I just feel good to be your winner, man. I tell you that. Two and two. Next up, I... Yeah, he's always on the cover. It's NBA 2K4, except it ain't really called. They call it the ESPN basketball. That's Iverson.    T-Mac, and nobody's correct me. All right, well, agent fans, he was wrong. It was AI the whole time. Oh, apparently. All right, give me that. Oh, I almost had it, too. I'm there, that's it. Cash, baby! Cash! And what? Cash! Hey, yo, Dirk is seven foot! He says that's horrible. He's out the truth, though. He's always been horrible.  Oh, no, sir! No, sir!  Let's go! I'm sweating. Just take a little bit. I'm sweating so much. Why would you stop running, you fool? No way. Does that mean you lose? No, no, no, no. You lose them, right? No, I hit the shot. I hit the first shot, bro. Oh, OK, OK, that's how I'm gonna play. Oh, yo, don't eat.  No, sir, no, sir! Game. Oh! Answer back! Getting awfully quiet there. No way, that's a fade. Yee-haw! You've done it and dunked it back. That was pretty hard, but you're still eliminated.  Next up, finally, we've got a different cover. It's Ben Wallace, NBA 2K5. Hold on, hold on, hold on. Are you sure about this one? I'm sure, bro. Like, bro, we're playing in Egypt, man! What else is there? I want to play in Egypt! No checks. We just play. No such thing as a check ball. Why they wearing track suits in this, bro? We're playing in Egypt, man.  All right, ice and motion. Ooh! Oh, that's buckets! I can't even see!  You got it back. How? I don't know. Is there no out of bounds? I don't think so.  Cash. No, sir. Answer back! Let's go. I can't even see that. Why is Dirk directly behind AI? That's my question. Oh, no. Man, you do this for a living.  Next up, we got Shaquille and Neil on NBA 2K6. You can have to win one of these, bro. I won two. I know, but it's on the Dreamcast, so it doesn't really count. You know what? Whatever. I won two.  Yeah, you should have this one. Wait, wait. You should have this in the microphone. Hold on. You should have this in the way. I'm actually sweating right now. Yeah. OK, you're sweating, too. Damn, I keep trying to tell people, man. I'm actually sweating. Oh, we got a whole little thing now! It's getting really serious, then. OK. Oh, you can't jump with square no more. It's triangle. OK, OK. OK, no, that's how we play now, though. Yeah, I'll keep terrible whatsoever. Well, this is game for you, if you miss. Bro, AI's doing this all by himself, bro. Oh, that's a two? That's a two. It doesn't count. I can verify this with Shaquille and Neil. That's Dwayne Wade. NBA 2K7. That's Dwayne Wade. It's nice. First of all, we're playing in China. That's amazing. 2K, how come we can't do that in 2020? He just said a screen. I'm pissed. I'm pissed. Oh, my God! What are you doing, you white-ass open? That's your favorite player right there. I'm right here with you.  That's a travel. I'm right there, please no. Ah, let's go! Answer back. Answer back. I just miss what I do. Answer back. Listen, man. This is ISO Joe we got right here, man. Please ISO. No sir! No sir! That's a dub, you feel me? It's been a long day. That's a dub. What game is that? I mean, sorry. Who's on the cover? Who's on the cover? What's the rules on cheating? Don't. Like, isn't that always the rule I feel like? Because if you're cheating, then it's clearly against the rules, so don't. You're too quiet. What are you planning? What the? I'm trying to pattern focus. Let me look. I'm right there, bro. If you... What did you do? How are you cheating? Let me see it real quick.  Let me see that. Let me see that. Please! Let me see it! I missed. You missed? You did, bro. I'm talking about it. Have a nice day. Ow! Come on. This game is banged up. This game is this banged up. I remember it. I remember it. Give me the ball. Let me have possession. If it doesn't score, you win. That's it. Oh, and it's Winner's Ball?  Give me the ball. Okay, so that's how you do that. Okay, hurry up. Because I don't know the controls, man. Hurry up.  I held it all the way. I held it all the way. Nah. So you quit. You're done. You lost. Is that better? Yes. He's going to hit that. This difficulty is a**, bro. It's so a**. Which one's the three? Why can't I switch players? Are we shooting NBA 3 or a quad? I can't switch players. Instead, he's jumping. Look at what Baron Davis is doing, bro. Please. I'm going with the severe depression, man.  This game is horrible. Do it. Do it. Hit the camera. Hit the camera. All right, so we didn't have NBA 2K10 for some odd reason, which is a tragedy, considering it's my favorite one. But here's 2K11. This one's better. It's not better. Darren Williams up, Chris Paul. Yeah, me. Ain't no controller going to stop me. Give me that. You know what I'm talking about? Give me that. Westbrook? No. Why did you do that? That's not how you cross over. Yeah, obviously not. Do you cross over? There's no such thing. Do you contest? Buckins. Get that out of here. Answer back. Answer back. No pump. Worst 2K of all time after 18 and potentially 20, depending on how this year goes. OK, this is where I leave, because I can't play that. Hey, man. How you doing? Doing good, June. You're like my talk show. I got a shot for the specialty shot. I just made a face like two minutes ago. I don't cut that out. What was the face you made? I don't know what you're on. Yo, what the? This room is so hot. I was perfectly fine up until four seconds ago. You hear that, guys? What? She was right, right? Right. You have to look down to go right. OK, yeah, I'm going to win. Wait, wait, wait. I don't think she would do that. He's not joking. He's not joking? He's not joking. All right, nice. Crossed out. Yeah. I don't know you're going to win. I just have a feeling. No idea what he's doing. He's making a sound. OK, where's the? I'm thoroughly depressed. I'm going through severe throat. Come on now. Back up, back up. Shoot, shoot, shoot. Oh, is that me? Yeah. That's so hard to tell, man. I don't know. There's names under the player. I think you have to hold it like way longer. You know what I mean? Just stand in front of him. Yo, move out my way. Just stand in front of him. Stand in front of him. If he misses, you win. I already lost. Oh, no, we both shot. We both shot red, so it doesn't count. I don't like a lot to you. We might just have to forfeit this one. I don't even know how to do it myself. How is this? How would none of these are green? How are none of these green right now? I have hope. I have hope. Yo, how did you do that behind the back? I'll show you. I didn't shoot that. I swear I didn't shoot that. I didn't shoot that. I just did. I did not click square. Run it. Next up, fellas, we got NBA 2K14. Wow, we're on the PS4 fun. We're playing on Hall of Fame difficulty, of course. And I believe the shot meter's on, so you have to get an excellent release or it does not count.  No! Please! I got one more chance though. Yeah, you got one more chance. OK, all right. I'd like Corey Joseph to shoot it too, if that's what you wanted. Pull up from deep. Very early. Yeah. Got that board, got that board, got that board. Next up, Kevin Durant, 2K15, ladies and gentlemen. What are you going to do? Nothing. Yes! I want to tip off. Because I wasn't looking. That wasn't a green. Please, where was it? No. Bro, these are not. You're just lying, bro. As I'm saying, I should have won. This, yo, does not make no sense. This is how I always. Why are the releases so slow? For you, Shabazz, not pure, he's in the sh. Yo, I'm pissed. I'm at this point, I'm furious. Bro, oh my god. That's, you mean to tell me that's not perfect?  We're in the fourth order. Let me open it. What? And this is like taking, this is a bit, this is actually a challenge. Oh! I was pressured. All right, here we go, baby. Can I have my way? I should give it to Willie, yeah. Because I think, what are you doing? Why did you do that? Why did you do that? You're welcome. Now you guys can move along. I won. All right. Okay, no, no, no. Teeth, teeth, don't. Watch it all fall out. Fall out. It's game. It's, yo, I'm, dude, that's my guy, Clay, bro. I tried telling you he had the best release. All right, you gotta answer back. I'm gonna come back to Caleb. Is Caleb on the court now? Let's go. Next up, 2K17 and the also infamous Paul George loading screen. The lead will be really happy with this one. Yeah, he'll be so happy. Very happy. I think he's probably throwing a party next week. 100%. I doubt it. You're even gonna fight it? This is it. This is it, fellas. Set the screen. He brings the ball up the court. Kairi Irving passes. Kairi Irving passes the ball. That's Paul George. Paul George brings it to the top of the viewpoint line. There's a double team. There's a double team. Paul George in the trap. I don't care! Escapes the trap. Bounce pass inside to Kairi at a bounce. Oh, yeah. Feels good to be a winner, man. I'm gonna tell you guys right now. You know what's crazy? I got you from my back here. Read it! Bam! Let's go. I'm getting bored of winning. I'm getting bored of winning.  Do the honors. Let me get the forms. Comfortable? Yup. Hey 2K19, Yanis is on the cover, fellas. No. Good stuff, Kevin Love. Beautiful trap. Hey, are you gonna agree on one of these eventually? Mr. Kevin Love? It's a wrap. It's a wrap. Get back, Uzi. Let me hit 2K20, guys. That was the closest I've been. Yeah, that was the closest you've been. That was the closest I've been. Why did you fade? I had a chance! Hey, thank you guys for watching.  If you enjoyed, drop a like. I'm gonna catch you guys in the next one. Thanks, Uzi, for joining me. Oh my god. Oh! What are you doing?  What are you doing there? Wear other shirts. Oh, Michelin S. I mean this. I don't know Michelin S. He did. He hurt his head. I know it. Thank you for watching the video.


We must communicate the effects of the cohesion policy in every day of our life, in my country, in my region, the big highways, the Museum of Acropolis or bridges and other big works. The people does not know and there are money from European community. We must do that. We must explain to the people what's happening every day with the Europe.


The latest date emergency management agency, LaSemma, has given an update on the collapse six-story building in Banana Island. LaSemma explained that the collapse was as a result of a truck that ran into the building. The Permanent Secretary of the Agency, Femmi Okia Osaintholo, confirmed that no life was lost in the incident, but recovery operations are still ongoing. The agency assured that efforts are ongoing to avoid a secondary re-enter into the collapsed site. We're moving swiftly, promptly, effectively and efficiently with all the Permanent stakeholders and you can see that immediately it occurred our response was swift, it's about ten minutes and you can see that come and look at the specialized equipment we used and we got there our response time was ten minutes, we moved in with our substituted equipment, we used Delta, the life side was divided into four cordons. Now, and we checked each of the cordons with our Delta and we did not, we did not locate anybody under the debris with our Delta. However, because we are professionals and meticulous about what we are doing, we are using a camera and excavator which cranes together. We started moving from one layer to another and during this emergency response we are able to rescue several people alive. Those people that were rescued were treated on the spot and discharged. It's only one that was taken to a private hospital where it's receiving adequate treatment. And if you can see that at the outer cordon, we've already reached what we call Grand Zero. We are moving to the second, third, fourth cordon and believe me, we will get to Grand Zero in each of these cordons to ensure transparency, to ensure that nobody is left in there, to ensure that the job is being done professionally concerning the other buildings. Hello, hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


Welcome to lecture 25 on measure and integration. In the previous lectures, we had started looking at measure and integration on product spaces. In the previous lecture, we defined the notion of product sigma algebra and today we will define the notion of product measure. .. So, let us recall. We will fix for today's discussion two measure spaces X, A, mu and YB mu. So, X is a set, A is a sigma algebra of subsets of X and mu is a measure defined on the sigma algebra A. Similarly, for the measure space YB mu, B is a sigma algebra of subsets of Y and mu is a measure on the sigma algebra B. So, we have already defined the notion of the product measure namely A cross B. So, if you recall, so we defined the notion of A times B. So, this is the sigma algebra generated by all rectangles and rectangles were defined as the sets A times B, where A belongs to the sigma algebra A and B belongs to the sigma algebra B. Now, we are given a measure mu on the sigma algebra A and given a measure mu on the sigma algebra B. So, that is a measure on B. So, our aim or the problem is to define a measure eta on the product sigma algebra A times B using the measure on A and using the measure mu on B. Why such things are important? So, let us just recall that on the real line and the Lebesgue measurable sets, we had defined the notion of the Lebesgue measure. So, that extended the notion of length. So, that extended the notion of length on R for subsets of R, which are not necessarily intervals. So, we want to do the corresponding thing on R 2. So, on R 2, given a set E, we would like to define the notion of area of E and if E is a nice set, for example, if E looks like a rectangle i cross j, then we know that its area, so let us call it as area of E is defined as length of i times length of j. So, this motivates the notion of the product. So, if we have sets in x cross y, so in general, if E is a set in x cross y and we have a notion of size here and notion of size here, then we will like to define the notion of size for subsets E in x cross y and for sets which are nice, which are very simple to describe, we will like to put it as a product of the length into breadth. So, the abstract question, the problem, abstract problem is the following that we want, given two measure spaces x a mu and y b mu, we want to construct a measure, let us call it as eta on the product sigma algebra a times b such that for sets which are rectangles, so what are the rectangles on abstract measure spaces, they are the sets of the type a times b, where a belongs to the sigma algebra a and b belongs to the sigma algebra b. So, for such sets, we want that the notion of the size for subsets namely, so our notion of size is the measure, so measure of a set a times b should look like mu of a, so something like length of a into mu of b length of the set b. So, this is the abstract problem, given two measure spaces x a mu and y b mu, how to define a measure in a nice way on the product sigma algebra such that on rectangles, it looks like the product of the corresponding measures, so eta times eta of a times b should look like mu of a times mu of b. In fact, this requirement that eta of a cross b is mu of a times mu of b itself says a way of doing this, so that means this fixes the notion of the measure for rectangles which are of the type a cross b. So, if we can show that this set function eta which is defined by this equation for measurable rectangles a times b by this equation and if you can show that is a measure it is countable additive, then we know that measurable rectangles form a semi algebra and they generate the sigma algebra a times b. So, we can take advantage of our extension theory and then extend this eta if it is a measure on the semi algebra of all rectangles to the sigma algebra a times b, so that is roughly the root we want to follow. So, to do that to implement this possibility, so let us write that eta defined on rectangles, so we are defining first eta on rectangles a times b, so eta of a cross b is defined as mu of a times mu of b. Obviously, it is a well defined set function and we want to claim that this is actually a measure, so eta of empty set is 0 that is because if a or b are empty set then this is 0. So, we want to show that it is a measure on R that means we have to show that eta is a countable additive set function. So, to show that let us take a rectangle a cross b a times b and suppose it can be represented as a union of rectangles a n cross b n which are pair wise disjoint. So, a cross b is written as union n equal to 1 to infinity of rectangles a n cross b n where all the sets a n's are all in the sigma algebra a the sets b and b n's are all in the sigma algebra b and these rectangles are pair wise disjoint that means a n cross b n intersection with some a m cross b m is empty whenever n is not equal to empty n is not equal to m. So, if a rectangle is written as a countable disjoint union of rectangles then we want to show that eta of a cross b is equal to summation n equal to 1 to infinity of eta a n cross b n. So, this is what we have to show. So, to show that let us proceed as follows. So, let us write, so here is a rectangle a cross b which is written as a union of rectangles a n cross b n n equal to 1 to infinity and these rectangles are pair wise disjoint. So, the disjointness we will represent it by putting a square cup. So, the notion of union instead of putting the u we will put it as square just to indicate that. So, we do not have to write every time that they are pair wise disjoint. The symbol itself indicates that they are pair wise disjoint. So, we want to compute eta of a cross b and show it is equal to summation of eta a n cross b n. So, to do that let us proceed as follows. Let us fix any point x belonging to a. So, for any x belonging to a, if we look at y belonging to b, then x comma y belongs to a cross b. We have fixed a and take any point y in b then the ordered pair x comma y belongs to a cross b. So, that is a cross b. So, it will belong to one of the sets here. So, which set it will belong? It will belong to some a n and b n. So, then x cross y belongs to a n and y belongs to b n. So, this implies there exist a n such that x comma y will belong to a n cross b n. So, that is a possibility. Now, if this weapon implies that x must belong to a n and y must belong to b n. If the ordered pair x comma y belongs to a n cross b n, then x belongs to a n and y belongs to b n. But what does that imply? That implies that if y belongs to b, that means implies. So, thus let us write what we have. y belonging to b implies y belongs to b n, where what is this n, where x belongs to a n. So, whenever x belongs to a n, y will belong to some b n. So, that implies that I can write the set b as union of our sets b n, where n belongs to, let me write S of x. So, what is S of x? S of x is the set of all those indices n such that it will belong to b n. That x, which is fixed, belongs to a n. So, out of the indices 1, 2, 3, so on, look at those n for which x belongs to a n. So, if x belongs to a n and then y will belong to some b n, that means y belongs to those b n such that x belongs to a n. So, this is what we want to claim. And not only that, we want to claim that these b n which are involved here, they are pair wise disjoint. That means this union is a pair wise disjoint union. Why is that? Because if this is not disjoint, that means if b n, if a point y belongs to b n intersection b n, where both n and m are in S x x, that means that will imply that x y belongs to a n cross b n and also it belongs to a m cross b m, where n and m are in the set. So, n and m are in the set of S of x. So, n and m belong to S of x. Suppose, I want to show this union is disjoint. Take two elements here, b n and b m. That means for n and m belong to S of x, look at the intersection. Suppose, there is a y in that intersection, that will mean what? That x belongs to b n and y belongs to a n and y belongs to b n. And similarly, x belongs to a m and y belongs to b m. That means x y belongs to both of this, which is a contradiction because a n and b m are disjoint. So, what we are saying is the following, namely we are saying the following that for any x fix, I can write my set b as a disjoint union of sets. So, this is what the conclusion is. I can write the set b as a disjoint union of sets b n's, which are coming in this union. But, which ends? Those ends such that x belongs to a n. So, to look at a pictorial view of this, let me just take a very simple example, illustration of this. So, this is the set a. This is the set a and this is the set b. So, we have got a rectangle a cross a cross b. We have got a rectangle a cross b and it is written as a disjoint union of rectangles. So, here are the disjoint union of rectangles. What are those rectangles? One is this rectangle. The second rectangle is this. The third rectangle is this. The fourth rectangle is this. The fifth rectangle is this and the sixth rectangle is this. So, this a cross b. So, the set a cross b is written as a disjoint union of six rectangles. So, let me call this first rectangle R 1. This is R 2. This is rectangle R 3. This is rectangle R 4, R 5, R 6 and R 7. So, these seven rectangles and their sides of each one of them, we can write down a 1 cross b 1. So, this rectangle, this side is a 1 and this side is b 1. For this rectangle, this side is this portion and this is the width and so on. Now, I wanted to illustrate that point. So, let us take a point x belonging to A fixed. So, when x is fixed, what are the points y in B n? So, to find those, let us go vertically. If we go vertically, for any y belonging to this set B, y belonging to B, either y will belong here or y will belong here. So, that means, this set B can be written as a disjoint union of this portion and this portion of B. So, that will be b 1 and this will be b 7. So, if x belongs to A 1, then y belongs can belong to B 1 or B 7 or it can belong to B 7. So, that means, B will be equal to B 1 union B 7. So, B 1 and union B 7. B 1 is the width, height of R 1 and B 7 is the height. For example, let us take a point x here. This is a point x. Let us take a point x here, which belongs to A 2. So, when I go above, for any, if I fix this, then how does the, how does the y split? So, if x belongs to A 2, if that is fixed, then to be inside the rectangle, I can be here, I can be here, I can be here. So, it will belong to B 2, B 6 and B 5. So, y can belong to B 2 or B 5 or B 6. So, that means, in that case, B will be equal to B 1, B 2, union B 5, union B 6. So, what we are saying? So, depending on where the point x is, the set n x will be 1 and 7. If x is in A 2, then n x will be 2, 5 and 6. So, B, in either case, B is a disjoint union of rectangles, some of the B i's. So, this is the important thing, which I wanted to convey. So, this is what is the conclusion of this argument, that if A cross B is a disjoint union of rectangles A n cross B n and I fix any point x belonging to A and analyze the points y in B, then x comma y belongs to A cross B. So, it will belong to some A n cross B n. So, y will belong to some B n's. So, which B n's it will belong? It will belong to only those B n's for which x belongs to A n. So, B can be written as a disjoint union of B n's, where n belongs to S of x. So, this is a disjoint union. So, that was the first important observation, once we have. So, once this is a disjoint union and all the B i's, the set B n's are all in the sigma algebra, where nu is defined. So, this being disjoint that implies, what we get is that nu of, so this star, so star implies namely that nu of B is equal to summation nu of B n's, where n belong to S of x. So, now I would like to transform this equation slightly. So, what was x? x was a point in A. It was, when x is in A, so for every x fix, that means for every x fix in A, we had this. Now, suppose if x belongs, so suppose x does not belong to A, if x does not belong to A, then obviously x does not belong to B n's. x does not belong to B n for every n, sorry. Then if x does not belong to A, then x does not belong to, sorry, not B n's, x does not belong to A n for every n. So, that implies that chi A n of x will be equal to 0. So, if x belongs to A, then x will belong, then it will belong to some of the A n's and in that case for those n, it will be equal to 1. So, if x does not belong to A and if x belongs to A and x, that means x will belong to, x will belong to some A n and that means this n will belong to nx and that means we will have chi A of x will be equal to 1. So, what I am saying is this equation, nu of B equal to this, I can write it as nu of B times the indicator function of A x is equal to summation over all n equal to 1 to infinity chi of A n x times nu of B n. So, let us understand this once again that why is this so. So, if x belongs to A, then the left hand side is this value of the indicator function is 1. So, the left hand side is nu of B, so that is here and if x belongs to A, then it will belong to some A n. So, if x belongs to A n, this value is 1, so the right hand term is nu of B n. If x does not belong to A n, then this value is going to be 0. If x does not belong to A n, then this value, so if x belongs to A and x belongs to A n, this value is 1, otherwise this value is 0. So, on the right hand side in this summation, only those terms will be non-zero for which x belongs to A n and that means the value of the right hand side terms will be indicator function will be 1 and nu of B n. So, this will be this equation, otherwise both sides are equal to 0, so that holds. So, what we are saying is that, so from the earlier equation, we have come down to the second conclusion namely this holds for every x belonging to A and so for every x belonging to x. So, for every x, this equation holds. So, that is what we have proved, because when x belongs to A, it is the earlier equation star. When x does not belong to A, both sides are equal to 0, so this equation holds. So, this is the second crucial step in the arguments namely, if A times B is a rectangle which is written as a countable disjoint union of rectangles A n and B n, then indicator function of A times x into nu of B is summation over n indicator function of A n times nu of B n. Now, this is the equation involving non-negative, this is the equation involving non-negative simple non-negative measurable functions, so on the measure space xA mu. So, this is a sequence, so now I can apply monotone convergence theorem. So, this apply monotone convergence theorem on xA mu. So, this is a non-negative function which is a limit of, so the sum means, it is a limit of the partial sums. So, it is a limit of non-negative measurable functions. So, monotone convergence theorem will give me that the integral of this is equal to limit of integrals of this. So, an application of monotone convergence theorem to this equation star gives me that integral chi of B indicator function of A x d mu x can be written as summation n equal to 1 to infinity integral of indicator function of chi A n nu of B n d mu x. So, this is a straight forward application of the monotone convergence theorem. Left hand side is a non-negative measurable function which is a limit of non-negative measurable functions on this measure space. So, integral of the left hand side with respect to mu must converge to the integral is equal to the limit of the integrals on the right hand side. So, this is an application of monotone convergence theorem. Now, let us compute the right hand side. The left hand side, the integral of nu of B, nu of B is a constant, so that goes out. Integral of the indicator function of A with respect to mu, so that is mu of A is equal to summation n equal to 1 to infinity and this integral again nu of B n is a constant. So, nu of B n goes out of the integral sign and integral of A n with respect to mu, so that is mu of A n. So, what we have gotten is nu of B into mu of A is summation nu of B n into mu of A n, but this thing is nothing but eta of A cross B and this is nothing but each term is nothing but eta of A n cross B n. So, what we get is eta of A cross B is equal to summation n equal to 1 to infinity eta of A n cross B n and that proves that the, so hence eta is countably additive. So, that proves that the eta is a measure. So, this is how one proves that eta which is defined, so eta which is defined as eta of a rectangle equal to A cross, rectangle A cross B to be mu of A into nu of B is countably additive. So, let me slowly go through the proof once again. There is only one small idea involved in it and other is rest is straightforward applications of the earlier results. So, let us write A cross B. So, I am going through the proof once again that A cross B is written as a countable disjoint union of rectangles A n cross B n and what we want to show that eta of the rectangle A cross B is equal to summation of the measures of the rectangle, each rectangle. So, summation over n of eta A n cross B n. So, to prove this what we do is as follows look at the set A cross B. So, fix any element x belonging to A, then for any y belonging to B we know that x comma y belongs to A cross B which is nothing but union of A n. So, that means x comma y will belong to exactly one of them, but which one of them. So, x comma y will belong to that A n comma B n whenever this x belongs to that A n because x comma y belonging to A n cross B n implies x must belong to A n and y must belong to B n. That means what we are saying is x comma y belongs to A cross B if and only if x belongs to A n and for that x, the y should belong to B n because x is fixed. So, that n is fixed. So, what are those n's which are fixed? So, y belongs to B n provided x belongs to A n. So, for a fixed x collect together those n's. So, find the set S of x all those indices n say that x belongs to A n. See, A n's are not disjoint. So, x can belong to more than one of the A n's. So, look at those if x belongs to A n, but for a fixed x it will belong to only one of them. So, if x belongs to A n then y will belong to B n. So, as x varies over A, for every fixed x you will get a collection of B n's. So, what are those B n's? Those B n's are indexed by n belonging to S of x say that x belongs to A n and this union is a disjoint union. So, for every x fixed in A we can decompose B into a disjoint union of B n's over those n's say that n belongs to S of x. This being a disjoint union because A n's B n's are disjoint we get our first inequality that for any fixed x in A nu of B is summation nu of B n's over those n's which belong to S of x. And now we observed that equivalently this thing we can write it as nu of B I can multiply it by the indicator function of A because x belongs to A. So, this will be equal to 1. And this nu of B n I can multiply if x belongs to A n that means n will belong to S of x. So, I can multiply here by the indicator function of A n if n belongs to S of x. And if x does not belong to A n that means it cannot belong to anyone of the A. So, this all the remaining terms here will be 0 and all the remaining this side is also equal to 0. So, what we are saying is for any x in A I can write this is equal to this and this equation makes sense whenever x does not belong to A also because if x does not belong to A this side is equal to 0 and that side x does not belong to A. So, it does not belong to anyone of the A n's. So, all the terms are 0. So, this equation first we can write it as indicator function of A times x nu of B is equal to summation of chi A n's. And now we realize that not only this equation is valid for x belonging to A this equation is valid for all x in x. So, once that is observed so that is what is observed here. So, what we get is that the equation chi of A x nu of B is equal to the summation chi of A n nu of B n for all x. And now this is the equation about non-negative measurable functions. So, left hand side is a non-negative measurable function which we can realize as a limit of non-negative measurable functions namely the partial sums of this series and apply monotone convergence theorem. So, that will give us that the integral of the left hand side is equal to summation of the. So, I can take the integral sign inside by monotone convergence theorem and say that integral of chi A x nu B d mu y is nothing but integral of this summation. And so here is the application of the monotone convergence theorem I can take this integral inside. So, that is equal to summation of integral of indicator functions. And now it is just a matter of writing down the values of this nu of B n is a constant so goes out. So, this integral is nothing but mu of A n so and that nu of B n. And the left hand side this was integral of chi A of A nu of B nu of B is a constant. So, that is integral of mu of chi of A with respect to mu so that is mu of A. So, that gives us that eta of A cross B is equal to summation eta of A n cross B n whenever A cross B is a disjoint union of rectangles. So, that proves that this is eta is a countably additive function. So, what we have gotten is eta is a countably additive function so let us just observe. So, what we got is we got eta defined on A cross B 0 to infinity by eta of A cross B equal to mu of A nu of B is a measure. We got that this is a measure on the semi algebra. So, this is important on the semi algebra A times B. So, implies by our general extension theory via outer measures and so on we can extend we can define eta tilde on A times B define eta a measure eta tilde a measure and eta tilde of A cross B to be equal to eta of A cross B. That means this eta can be extended via outer measures to the sigma algebra generated by A cross B the semi algebra A cross B. And if you recall we had said that this extension will be unique provided this eta is a sigma finite measure. So, we claim eta is sigma finite if mu and nu are sigma finite. So, we want to show next that if A and B if mu and nu are sigma finite. So, let us assume so if mu sigma finite so that implies I can write x as a disjoint union of sets x i 1 to infinity each x i in the sigma algebra A and mu of x i finite for every i. And similarly, nu sigma finite implies I can write y as disjoint union of sets y j, where each y j is an element in the sigma algebra B and nu of B j is finite. And then this implies we can write x cross y as disjoint union of x i cross disjoint union of y j's. And now it is just a simple matter to set the equality namely this is same as the unions over i unions over j of rectangles x i cross y j. Because if x comma y belongs here that means x belongs to the union x i's and y belongs to union y j's. So, that means x belong to only one of x i and only to one of y j's. So, it will belong here and conversely. So, this is a disjoint union and now we only have to observe the fact that x cross y has been decomposed into a disjoint union of sets x i cross y j and we only note that eta of x i cross y j it is a rectangle. So, its measure is mu of x i times mu of y j and both of them being finite. So, this is a finite quantity. So, x cross y is written as a disjoint union of sets x i cross y j and each piece has got finite measure. So, that implies eta is sigma finite. So, the measure eta is sigma finite on the rectangles and hence has a unique extension to the sigma algebra. So, this is what we wanted to prove that eta that extension is also sigma finite. So, general extension theory gives me a unique. So, that mu and mu are sigma finite. So, that implies x is a disjoint union y is a disjoint union. So, we can write x cross y as a disjoint union of the rectangles x i cross y j. That is what I just now illustrated and each piece has got a finite measure. So, by that process we get a eta is sigma finite on rectangles. So, by extension theory eta can be extended uniquely. So, that is the important thing eta can be extended uniquely to a measure on a on the product sigma algebra. So, that for rectangles it is the product. So, this is the measure eta which is defined on a times b on the product sigma algebra is called the product measure and is normally denoted by mu cross mu. So, let us summarize what we have done. We started with the two measure spaces x a mu and y b mu and for the product set x cross y we first define the rectangles namely sets of the type a cross b where a belongs to the sigma algebra a and y belongs to the sigma algebra b. So, that gives us sets of subsets of x cross y called measurable rectangles they only form a semi-algebra. So, we extend here, we generate the sigma algebra by this semi-algebra of rectangles and call that as the product sigma algebra denoted by a with circle cross a times b and now given measures mu on the sigma algebra a and a measure mu on the sigma algebra b we want to define a measure on the product sigma algebra. So, that is done by defining the product for a defining the new measure first on rectangles. So, eta of the rectangle a cross b is defined as the product of mu of a and mu of b and we show that this is a measure. So, this becomes a measure on the semi-algebra of rectangles and if it is sigma finite that means if we assume that the given measures mu and mu are sigma finite then this extends uniquely to a measure on the product sigma algebra a cross b and that measure is called the product measure product of the measures mu and mu. So, given two measure spaces x a mu and y b mu which are sigma finite we get the product measure space x cross y the sigma algebra a cross b generated by the rectangles and the product measure mu cross mu go obtained via the extension theory. So, this is the product measure space constructed as just now said. Now, the next problem we want to analyze is the following namely this product measure mu cross mu that we have gotten is obtained via extension theory, but it does not tell us how does one compute the product measure mu cross mu of a set in a cross b. So, that is not indicated because we are making use of the extension theory. So, next problem that we want to analyze is the following. So, namely so we have got the product measure space x cross y the product sigma algebra a times b and the product measure mu cross mu. So, let us take a set E contained in x cross y which is of course E is an element in a times b. So, mu cross mu of this set E is defined. So, the question is can we compute this quantity mu cross mu of E using mu and mu? So, that is the question. And there let us just recall something from our elementary calculus. Supposing in the plane we have got a set which looks like the following. It looks like this is a set. So, this is a set E which looks like the following namely here is a point A and here is a point B. So, the set E looks like, so let us just write what does the E look like? E is equal to all x cross x comma y so that x belongs x belongs between A and B and y. So, at any point x if I look at y this is the portion of y. So, it starts with the green boundary. So, y is bigger than or equal to some function f of x that is the green curve and less than or equal to here is g of x. So, this is what we call in calculus or elementary analysis sets of type 1. And for such sets for such sets we can find out what is the area. So, area of the set E if you recall from calculus it can be obtained as you look at this difference height what is this height. So, that is nothing but g of x minus f of x and integrate that from A to B dx. So, Riemann integral as an application of Riemann integration we do that we define it equal to this. But now let us rewrite this. This I can write it as this is Riemann integral. So, Riemann integral I can write integral over AB of D lambda with respect to the Lebesgue measure. And what is g x minus f x that is precisely the Lebesgue measure of this Lebesgue measure of this height. So, Lebesgue measure of let me write as E x what is E x? E x is equal to all y such that x comma y belongs to E which is same as all y such that y is between f x and g x. So, that set I am writing it as follows. So, I am writing as Lebesgue measure of a notation called E x. So, you can think of that look at this set x let look at that look at that set E to find its area we are just adding up the areas of these small strips. So, I can think it as that way that is what this integral seems to indicate. So, we would like to generalize this in the case of our construction the same idea we want to generalize it. So, here is what we want to do. So, given a set E in x cross y for x belonging to x fix let us look at E x that is so here is abstract now x is abstract set y is some abstract set. So, look at all those points y belonging to y section is the part of the horizontal line and the x section is part of the vertical line. So, as I said E x is called the section of E at x or just the x section of E and similarly E y this set E y is called the section of E at y or just the y section of E. So, here are some simple properties we want to verify for this sections. So, first of all we want to verify and let us look at some examples first. Let us take a set E which actually looks like a rectangle. So, in this x cross y let us take actually a rectangle A cross B where A belongs to A and B belongs to B. So, then for any x in A if for what are the points y such that x comma y will belong to E that is means y must belong to B. So, E x the x section of E for a rectangle A cross B is nothing but B the set B itself if x belongs to A and if x does not belong to A then the point x comma y is never going to belong to E. So, the x section is empty set. So, here is a simple observation that for a rectangle A cross B the x section is equal to the set B if x belongs to A and it is empty set if x does not belong to A. Similarly, the y section of E or the section of y at a point y in y. So, all x say that x comma y belongs to B then for all x in A x comma y is going to belong to E. So, that means the y section of E is equal to A if y belongs to B and it is empty set if y does not belong to E. So, for rectangles these are very easy to compute what are the sections. For a rectangle A cross B the x section for x belonging to A is B otherwise empty. Similarly, the y section is equal to A if y belongs to B otherwise it is empty. Now, let us look at another example. So, let us take a measurable space x A and look at the ordered pairs x comma t. So, t belongs to R such that this t lies between the evaluated indicator function of A at x. So, we are looking at the ordered pairs x comma t such that for every x t lies between 0 and A and x belongs to x. So, what are the sections of this set E? This is a subset of A cross B and where B is y is the real line. So, it is subset of x cross R we want to find its sections. Let us observe that for a point x in A if x belongs to A then this indicator function of A the value will be equal to 1. So, t will be between 0 and 1. So, if x belongs to A then t will be between 0 and 1. So, the section is going to be the interval 0, 1 and 1 not included and if x does not belong to A if x does not belong to A then this is going to be 0. So, t is going to be the singleton 0. So, the section if x belongs to A so the section depends on whether x belongs to A or not. So, the section of E at a point x is equal to the interval close interval 0 open at 1 in R if x belongs to A otherwise it is the 0 set or another way of looking at this is the following that the set E I can write it as A cross the interval open close at 0 and open at 1 union A complement of this A complement cross the singleton 0. This is another way of writing the same set E as I explained just now. So, the section now is a union of two disjoint rectangles. So, section in the first case when x belongs to A section is going to be 0, 1 and in the second case the section is going to be the singleton 0 if x does not belong to A. So, these are the x sections. We can similarly find the y sections. So, for y belonging to 0 to 1 that means y is the real line. So, for a real number between the closed at 0 and open at 1 interval it is going to be A at 0. So, and if y is equal to 0 then this is going to be the whole space x and empty set. So, this is easy computation from this it follows. So, this is how one computes the sections of these sets. So, these sections are going to play important role in computing the measure of a set E in the product space. So, in the next lecture we will analyze the x sections, the y sections, various properties of these sections under complements, intersections and unions and then show that each section for a set E in the product sigma algebra, each section is again a appropriately measurable set whose measure can be defined and then you can take the measure of that and define the functions and compute the integral of the product set, compute the product measure of the set E. So, we will continue this study of sections and their implications for product measures in the next lecture. Thank you.