Welcome to NTN Nightly, I am Anisia Antoine, this edition's top stories. Tourism minister Nabil Dominic Fede to hold talks with stakeholders on the performance of the tourism industry. Artist Gary Butt pays homage to the community heroes of Marshal. And the inaugural Urban Adventures summer camp sets participants on the right path. The Government of St. Lucia has welcomed the recent Centers for Disease Control, CDC, upgrading of the island to level 1. This is a credit to the protocols that were put in place to address the pandemic, the leadership of the Prime Minister, the Cabinet of Ministers, the Ministry of Health, the Chief Medical Officer, and all St. Lucians who continue to adhere to the protocols. Speaking about the recent CDC upgrade, St. Lucia's ambassador to the United States, His Excellency Anton Edmonds, says the CDC report is but one input that informs the state's department's travel advisories. His office, he says, is continuing to engage the relevant authorities, advising and updating them on the protocols that position St. Lucia as a location where safety is a priority of the government for citizens and visitors. Prime Minister Hon. Alan Chastney also continues to advocate on St. Lucia's behalf and has been a champion on the matter of protesting for travelers to St. Lucia and the wider Caribbean. The Prime Minister noted that the success the country has experienced in the management of COVID-19 is in large part due to the cooperation of the population and the hard work of the team at the Ministry of Health. Meanwhile, St. Lucia has also been singled out by the AOL News as a safe place to wait out the pandemic. The selection has generated interest in the island which tourism minister Hon. Dominic Fede is hoping to convert into actual arrival figures. Lisa Joseph reports. Tourism Minister Hon. Dominic Fede says St. Lucia has undoubtedly done well on containing the novel coronavirus and the upgrading of the island by the U.S. Centers for Disease Control CDC to level one, meaning low COVID-19 risk, is an acknowledgement by the international community of the hard work done by the government and people of St. Lucia. Despite the high ratings, the tourism industry is on a slow climb with market performance remaining low. Load factors for the airlines are averaging 30 percent on the destination route. Hotel occupancies are also low affecting profitability. Hon. Dominic Fede says the Ministry and stakeholders are assessing, reorganizing and improving the tourism products. We have an opportunity with the recent announcement by the CDC that St. Lucia is one of the safest places in the world for people who are looking to wait out the COVID pandemic to travel to and to go to. We certainly are number one in the Caribbean from all the reports that we have seen. We are the best country in terms of how we have managed COVID-19 in the Americas or the Western Hemisphere. So we are very, very pleased that we are among one of only eight countries in the world by the CDC. That is quite special to be named in such a prestigious and elite group of countries that have really done an exceptional job in management of COVID-19. The Ministry is continuing to dialogue with the various sub-sectors of the tourism industry. Governments were held this week with car rental operators, events and promoters to find the best and safest possible way of reopening. Minister Fede says government is deeply concerned about the thousands of St. Lucia's who have been made jobless due to the impact of COVID-19. We are going to be talking to the hotels a little later this week to see what we can do, how we can work together to make it a little better for them. All of the sectors that we have spoken to are having significant difficulty whether it's events, whether it's the day boat operators, whether it is the car rental companies, which I just came out of the meeting. It really isn't a rosy economic picture for these sectors. And we are working very feverishly to see how we can create the best able environment for them, even amidst this very difficult time. An estimated 2,000 tourism workers are back on the job in the hospitality and taxi sectors. From the Government Information Service, Lisa Joseph reporting. 15 tourism properties meantime have been COVID-19 certified for reopening. Tensin sectors like sites and attractions and dive activities are working towards resuming. Donaldine Vita is the permanent secretary in the Ministry of Tourism. We continue to look at what the probability of risk is and at what time we could determine what services that can be added to the pool. Dive is an activity that can happen very safely and we have been able to work with the Dive Association and the many dive operators. There are dive operators at the hotels that are open as well. And as of Monday gone, 17th of August, we had the resumption of dive activities that really joined the day boat activities that we had started. Apart from dive, we are also considering and working with sites and attractions so that day two can get themselves ready. The protocols that they have to put in place, they have been open for the local public. However, when we start to look at the resumption of service to the high-risk market and the COVID-19 certified properties, we want to ensure that they go top notch in their cautions and their controls. So we're working with them to ensure each and every one of them who have presented themselves to participate in activities to their plan, they are ready and they're sanctioned to operate. A mural project is now underway in Marsha Castries to commemorate the heroes of that area and draw awareness to their significant contribution to community development. Jesse Laos tells us more. Visual artist Garibute is busy transforming the empty out walls of the Mindu Philip Park into an ode to outstanding citizens of the Marsha area. It is a mural project that depicts the likeness of these community heroes, ensuring that current and future generations know of their legacy. Viewt listed some of those featured on what has now been dubbed the hero's wall. We have Alicia John who is the principal in the JP in Marsha. We have Pan Andrew Shining who has made this contribution in Steelpan. We have Joyce Ogis who is a cultural icon. She has done a lot in sports as well. We have Arnold Cloudon who has done a lot in sports, in football, in coaching the younger generation and I think that he should also get some kind of recognition. We also have James Belgrave who is another one. In fact, James Belgrave has a court inside the Mindu Philip Park that he made his own contribution to see that the sports fraternity develop back in those days. Through the visual aesthetic, murals are said to promote a sense of belonging and strengthen community identity. From the onset, there was engagement with Marsha residents on the figure's best suited for this recognition. This project has been going on for a period of about two months. It really started with conceptualizing, doing research, engaging the school. I must say that the Marsha Primary School has really been involved in the research with the teachers and students, getting information on the people of the community and getting a lot of feedback from the people as to who is best to be on the wall. Though incomplete, the hero's wall has already become a conversation starter with residents who are eager to learn more about the individuals being painted. The muralist hopes for versions of this type of commemoration to be recreated in other communities island-wide. I think it's something that can also be replicated in many other communities because I am very, very certain that a lot of communities have hidden heroes that people do not know of. The only way you can get to know about them is to put them out there. And this is why I will say it again that we have come up with that concept for people to recognize their heroes and see how much that they have contributed and how now the younger generation could follow that pattern. Meantime, the Marsha project is still in its first phase, and unveiling of the finished mural will be held upon its completion. This East Castries mural is the latest undertaking by Gary Butte, who has over 20 years of artistry under his belt. For the Government Information Service, I am Chessie Léonce reporting. More than 20 young persons are better equipped to explore entrepreneurship after participating in the inaugural Urban Adventures Summer Camp. The Junior Achievement Program of St. Lucia hosted the closing ceremony for the first-ever Urban Adventures Summer Camp. The Junior Achievement Program focuses on three educational pillars for success, financial literacy, entrepreneurship and readiness. The camp focused on the new strategies of the Junior Achievement Program and created an avenue to develop the youth and ensure sustainability for the program. Approximately 21 participants graduated from the two-week-long program. Parliamentary representative for Grosjele Honourable Leonard Montoute noted that the different aspects taught at the camp will aid in the holistic development of the youth. These are qualities, these are skills that you should begin to practice as of now and take with you for the rest of your life. To learn to become an entrepreneur is more important than ever. Gone are the days when you can have an education in St. Lucia and expect to get a job in the public service or in some establishment. More and more, you have to begin to think innovatively as to how you can provide employment for yourselves. That will become very critical in the aftermath, as if I know that COVID-19 is about to end. But whether we have to live with it or if we are fortunate enough anytime soon to come out of it, I know that in the aftermath we will need the qualities that you've learned here today. Chairman of the Junior Achievement Program, Kurt Hosam, expressed gratitude to all stakeholders involved in the initiative. It is said it takes a community to raise a child. So I found it fitting that one of the, apart from etiquette fundamentals of banking and finance, arts and crafts, teamwork, savings and physical fitness that are all necessary, we dealt with agriculture, the fundamentals of agriculture. And I want to draw parallel to us today, that we have a responsibility to continue to sow seeds for the future in our children today. Everybody could agree with me? Yeah? We are sowing seeds. This program is sowing seeds of development for future, for our future. Because this is the future of the nation. It is within our young people. And in preparing them with such fundamental skills, enables them to be able to go out and to be functional in our world. And that is at the heart of Junior Achievement. We want to encourage the spirit of entrepreneurship and we want to do so in a collaborative fashion. Jim Zavier, Director of Community Service, encouraged participants to remain disciplined and keep focused on their goals. I want to admonish you to continue in this vein. You are on the right track in life. That with discipline and the level of focus that you've shown that you are destined to reach very far. I want to compliment the parents. You've done an excellent job. Continue parents providing the support for them. The ministry will be there to support JAA and the other community organizations to ensure that they go very far. The closing ceremony for the Urban Adventures summer camp took place on Wednesday, August 26, 2020. From the Government Information Service, I am Anisia Antoine reporting. This is NTN Nightly. Up next, Primus Hutchinson with the NTN Novella Quay All. In an effort to ensure patient and first responder safety, the St. Lucia Fire Service has reviewed its patient transfer procedures, especially for patients with respiratory distress. Face masks will be provided. At no time during transportation should the face mask be removed. Please be patient and cooperative during this time to ensure you receive the best possible care while keeping our first responders safe. Welcome back. We join Primus Hutchinson for the NTN Novella Quay All. Thank you, Anisia. Thank you, Madam Department of University of Saveleté for information on government services, GIS and the NTN National Television and the NTN Novella Quay All, Primus Hutchinson. As an artist, I have an initiative to serve as the director of Common Marcha, which I have made many contributions not only to Common, but also to the general public. Garibut, the director of your mural, has made many contributions to music, sports, and education at the same time. Garibut, I work in a way to encourage young people to appreciate the contribution of the biggest citizens for their development, especially here in total. I declare that the mural is a gift for Common. It is a gift for Common, for people to appreciate. It is a gift for the people who are very generous and are willing to do what they can. It is a gift for Common, for people who are very willing to do what they can. So, I think it is a gift for Common, for people who are very generous and are willing to do what they can. Because now, everyone is like Common. Now, everyone is like Sophia, Gauzile. We are like Common, but Common is a gift for the people who are willing to do what they can. But we are like Common, we are willing to do what we can. So, Common is a gift for Common, for people who are willing to do what they can. Common is a gift for the generation who come here. So, we can connect with them. We can take place with them. Mr. Barosi mentioned the dance of Gwekla, which is performed in Sugwamiro. I am Alishatron. I have been a part of the community. I am Pan Andrew, I am a shining. I have made a music still Pan for Common. We are like James Bell Grave, so that I can help people who are young generation. We are like Al-Claude, so everyone who is a football player, who is a coach, all these young people, all these young people. We are like Joyce O'Giz, who is like Common, who is like a football player. So, we are like a few other young people. We are like a few people who come here because we are like Telma. We are like Telma, we are like the capital of Kastui. This week, the Ministry of Health continued to take care of the public because it is the pride of Gwekla that is growing here. It is also the position of Gwekla, because it is the pride of Gwekla that is growing here. It means that the world that I have been able to affect, the people that I have been able to help for the present, for the future. And I have been very necessary for the Ministry of Health to take care of the situation immediately. According to epidemiologist Dr. Michel François, this is a qualified doctor for a medicine. He told me that since the week of the 15th, I have been able to record and confirm that the fever has gone up to 168 cases. He said that despite the fact that there is still death, as a result of the fever, the majority of these cases are in the age of 6 or 13. Dr. François declared that he had discovered that all of the fever cases are in this situation. Particularly in the north, in the south, and in the south, and in the south, and in the north, and in the south. This is a very concerning situation, because the situation is that it is a place where you can no longer afford to pay as you have been fighting despite the corona virus. Particularly in the north, and in the south, it has also affected the economy, it has also affected the health, it has also affected the tourist. This is a very concerning situation, because it is a place where you can no longer afford to pay as you have been fighting despite the corona virus. This is a very concerning situation, because it has also affected the health, this is a place where you can no longer afford to pay as you have been fighting despite the corona virus. This is a place where you can no longer afford to pay as you have been fighting despite the corona virus. This is a place where you can no longer afford to pay as you have been fighting despite the corona virus. This is a place where you can no longer afford to pay as you have been fighting despite the corona virus. Gov. Settlesie, I welcome you to come out of this hand. Kamutwe Settlesie is at the bottom of his list of countries that have 8 coronal diseases. This information came out of Settlesie, which is what it is for the control of diseases in the US. For this reason, Kamutwe Degui, the effort I have made by Prime Minister Settlesie, we have Alain Chassney, the cabinet's mums, the Gov. Settlesie, the Chief Officer of the US, Dr. Sharon Belwood George, and all Settlesie who will continue to follow and obey these protocols that apply to Settlesie. Ambassador Settlesie from the United States, Anton Edmonds, said that if the United States can change the power of Settlesie, it will not be possible to change the power of Settlesie. As a result, Ambassador Edmonds said that he will continue to engage with the authorities and to advise them on how to control Settlesie, how to protect Settlesie and its citizens. Prime Minister Alain Chassney said that he will continue to seek support for Settlesie, for Settlesie, and for all of us who will be able to test, who will be able to travel around the world with Settlesie. Prime Minister Chassney said that if this is the case, and that Settlesie can change the power of Settlesie, it will not be possible to change the power of Settlesie. As a result of cooperation between the people of Settlesie and the work of the Ministry of Health, Prime Minister Tiki said that this is a good news for the people of Settlesie. It is necessary for Settlesie to continue to follow all the protocols and to adopt the habit of Settlesie, so that the people of Settlesie will never lose their attention and that Settlesie will be able to control the power of Settlesie. That is why I would like to thank the Minister of Health for the invitation. I would also like to thank the Minister of Health for the invitation. You can also catch up with us anytime on the St. Lucia Government Facebook page or YouTube channel. I am Anicia Antoine.


What's up guys Jay here reporting to you again from comic get in TV. I'm here with something really special today It is the Walmart exclusive black panther Action figure Marvel Legends very cool very awesome Pick this up a couple days ago seen him. It was just like Gotta get him. He is amazing If you don't know much about black panther movies been out for like a while now Possibly a month and it's dominating like number one movie since it's been out This thing is huge He made his live-action debut in Captain America Civil War people went nuts Like completely nuts. He was so awesome He'll also be in the Avengers Infinity War movie coming out next month April 27th very awesome can't wait so excited for this movie.  I promise check this thing out Alright guys here he is as you can see He is actually pretty good looking. I'm enjoying him. He comes with an extra head Which is all black These go on to the fist hands Which I already have one on He comes with an extra set of the fist and the claw which I enjoy the claw better because that's where like You know Black panther He bends pretty easily. I'm definitely enjoying them That's pretty nice Definitely stands really nice Definitely a cool figure.  I Love I'm loving.  He's got some on his back Um Definitely a cool figure the knees bend rather easily Feet bend pretty easily arms are definitely nice So that's cool. He's you know set him up to look like he's in panther mode like he's about to attack the heads nice Seems it might be a little hard to take off which isn't too bad, but I'm enjoying it. I like the purple head though. I don't know how I'd feel about putting on the Blackhead With the suit, you know what I mean? I'm just I'm feeling these lines. It's kind of like he's Striped or something. It's really cool. I'm definitely digging this figure He's probably one of my favorite Marvel legends that I've seen in quite a while Other than that, I'm definitely going to give this figure like ten out of ten just because He moves really well And he just looks super cool like I just can't get over how well this guy turned out, you know But definitely definitely cool well guys if you liked this video hit like down at the bottom don't forget to subscribe and Check back next time for another episode only on comic get in TV. You guys have a great night stay geeky


Zgledaj. Tako očasno, naredaj smo počasno odvihčili vseh, da je zavrčil posponil vseh odvihčili. Zato priberimo, da je to priberila. Zatimamo, da je zatečnjamo zbičnjak vzelo in v 0-1 vzelo v 0-1 vzelo, da je zato vzelo. Ok, skupnja. Ok, f is continuous is increasing. Oh, non decreasing actually. Ok, it's continuous because we saw that it is a uniform limit of continuous function. Ok, we saw that f is 0 is 0, f of 1 is 1, and we saw that it is, by definition, it is constant on the interval, on the open interval that we remove in the construction of the counter function, ok? On the counter section. Ok, now by means of the counter function we can construct anemomorphism between 0, 1 and 0, 2. Ok, so let us consider h, we define it by h, the following function between 0, 1 and 0, 2, such that 2x associate x plus f of x, where f is the counter function, ok? Ok, so we immediately saw that h is continuous because to continuous function is strictly increasing, ok? So h is continuous, strictly increasing, and so in other words h is anemomorphism between 0, 1 and 0, 2, ok? Ok, so somehow the final purpose of this first part of the lecture would be to see that the notion of measure is not preserved under anemomorphism, ok? Ok, so we will see that h can map a set of measure, a measure 0 into a set which has a positive measure, so the notion of measure is not preserved under anemomorphism, we will need to require more in order to preserve a measure. Ok, so consider, ok. Consider k is the counter set, the counter set. Ok, then we have that the complement of the counter set is the union of this opening interval that we removed in the construction, ok? So is the union just to recall this joint interval. Ok, ok, we know that. As I recall you before, we know that f, the counter function is constant on each such an interval, ok? Ok, so by construction the counter function is constant on each i n, and so, what about? We have that, this means that for any x in this open this joint interval i n, we have that h of x, and for omeomorphism what looks like as x plus some constant c n, ok? Of course the constant depends on the interval. Ok, so what is the image under h of i n? Ok, so we have that h i n would be i n plus c n. We have that the measure of h i n is the measure of i n, ok? Because we saw that the measure is invariant under translation. Ok, ok, just let me call this equality star. Ok, we know that since h is omeomorphism and so it's 1 to 1, we have that, we have that the image of the union of this joint set will be the sum, the union of the image of the set. Ok, we have that. The union, the image of the joint interval is the union of the images of the interval, ok? Of the images, and they remain this joint, ok? And they remain. Ok, so you have that. Ok, call this complement a. So you have that h of a is equal to the union, countable union of h of i n, ok? Ok, now we want to compute the measure of h of a. So we have that the measure of h of a is, we use this, is the measure of the union of h i n, which is equal, because we saw that they are disjoint, it's the sum of the measure of h i n. Ok, this is because of this remark, and this is because of star. Ok, so this is equal to the sum of the measure of i n, which is equal to the measure of a, that we saw on Tuesday that this is equal to 1, ok? Ok, so again we use the fact that h is 1 to 1, it's a normal morphism, so we have that. Ok, again, so since h is a morphism, ok, the complement, ok, the complement of the image is the image of the complement. Ok, so in particular we have that, we consider h of k, where k is the contour set, this will be equal to 0,2 minus h of a. Ok, so we compute h of k, the measure of h of k is equal to 2 minus 1, which is equal to 1, and so we saw that finally what we prove is that h map a set of measure 0, which is the contour set, ok, into a set of measure 1. So basically this is to say that the notion of measure, of the bag of measures in this way is not topological invariant, ok? So it's not preserved under homomorphism. Ok, now we start with another argument, with a new argument. Ok, so the next argument will be the measurable function, ok? Can you see? Ok, so before introducing the definition, I would like just to anticipate you that when we will introduce the notion of the bag integral, we will have to deal with the set of, sorry, the bag integral for function f, for instance, we will have to deal with some function f, put like this. So we will have to deal with this kind of set, ok? And so somehow we would like to know, to classify, to characterize the function f for which this kind of set is measurable, ok? Otherwise it won't make sense to compute the measure of this kind of set. Ok, so from this somehow remark, we define what is a measurable function, ok? So consider a set A, which is a subset of R, of course, and the function f defined on A, which values in the extended R, so which is the set of the extended real number, so you have R if you want union minus infinity and plus infinity. So we will say that this function f is measurable if two facts are fulfilled, so is measurable, ok, right here, if the domain R is a measurable set, is measurable to I, if you have that for any alpha in R, the set f larger than alpha is measurable, ok? So two, somehow there are two requirements. Ok, so from this you can see that if this set is measurable, also there are many other set that we can infer are measurable, so we see the following proposition. So let f be defined from A to R be a measurable function, and, ok, so of course it's already included in the definition, but just to stress that we already know that A is in R, ok? I mean, ok, this is the set of the x in A, the domain, such that f of x is larger than alpha, ok? Ok, in the following for sure I will denote this set f larger than alpha, ok? Just to be, but, ok. Ok, so we have that, the following, if you start from this, we have that the following statement are equivalent, ok? Ok, then the following statement are equivalent, ok? So we have I for any alpha in R, ok, we start by this, we have that, ok, x in A such that f of x is larger than alpha is measurable, ok? Then we have 2I, ok? So again for any alpha in R, the set of the x in A such that f of x is larger or equal to alpha is measurable, and then we consider the other inequalities, 3I for any alpha in R, the set of x in A such that f of x is less, strictly less than alpha, it's measurable, and finally, for any alpha in R, the set of the x in A such that f of x is less equal than alpha is measurable, ok? Ok, so we start by proving that I implies 4, which is the easy case, it's very easy sometimes, because just you consider the complement and you are done. I don't get your, in the definition you mean, the proposition. So we start by f be a measure of a function, which means that it satisfies these two, it's a function from A to the extended real number, and by definition we know that A is measurable, I write it again, but it was not, no, no, but we are under the hypothesis that f is measurable. Ah, I mean, this is already, ok, we can remove it, ok, or, this is already included, ok, I know what you mean. Ok, let be a function, ok, thank you, yeah, sure. So let be a function, ok, we require this, because if we don't require it, it's measurable, we have to require it, yeah, sure, I agree with you. Then the following statements are equivalent, ok, so if you read this I mean that it is measurable. Ok, yeah, sure. Ok, so, ok, so we start by proving that I implies the last one, ok, so we can just see the set, ok, x in A, such that f of x is less or equal to alpha as a minus the set x in A as f of x is larger than alpha. Ok, so we know that this is measurable and the complement is measurable as well. Ok, then ok, for i is implies i is similar, you can argue similarly, now it's equivalent by exercise if you want.  Ok, and then 3i implies 2i it's a similar, it's measurable so the complement is must also be measurable and so it remains to prove that i for instance i implies 2i ok, we can write this set here as the intersection for instance so you have x in A such that f of x is larger or equal than alpha has the countable intersection of x in A such that f of x is strictly larger than alpha minus 1 over n, ok? Ok, so we know that these are measurable and the countable intersection of measurable set is still measurable ok, so this concludes the proof ok, so ok, now we want to deal with the level set of measurable function ok and we prove the following ok, so let f from i to r a measurable function ok, then we have that for any alpha belonging to the extended real line so it might be also plus or minus infinity we have that ok, the level set of f so the point in A such that f of x is equal to alpha is also measurable ok, we divide two cases the cases in which alpha belongs to to r so this is finite so you can see it as an intersection of two former set ok, so you have that this case x in A such that f of x is equal to alpha you can see it as the x in A such that f of x is larger or equal than alpha intersected the set x in A such that f of x is less less or equal than alpha ok ok, then the case when alpha is for instance plus infinity you need to do countable intersection so if alpha for instance is equal to plus infinity so you have that avoid the x in A such that f of x is equal to plus infinity intersection over n of the set x in A such that f of x is larger or equal than n ok in infinity you intersect the same so you have f of x less or equal to minus n ok, so this concludes the proof ok, now we give definition of measurable function but we would like also to somehow to see some example of how a measurable function look like ok so think about a continuous function so some example of measurable functions ok, so consider f from A to in n the domain is a measurable set and assume that f is continuous ok and see that f is measurable so do you see you know why the preimage of the set like alpha plus infinity is an open set so the preimage is open so you take the intersection with A it will be open the relative topology so you have actually an intersection of two measurable set which is measurable ok, so just you have that so we want to establish that it is a measurable set you can see it as the preimage under f of alpha plus infinity ok and this is open this is open so the preimage is open is an open set in the relative topology and it is open set of A if you want which means that it exists in U op and I said U open of R such that this set can be view as U intersected A U is open so it is a borrow set we saw that borrow set are measurable set and this is this is measurable so we get that also this set is measurable ok, so this is one example of measurable function then you can construct also another example you start by a set A in R and ok, I don't know if you know what is the characteristic function of A is a function from R to 01 ok, you can take for instance 2 to the full R so this is called characteristic function of the set A and it is defined in this way so you have that set A of X is equal to 1 if X is in A 0 otherwise so actually it takes 2 values ok, so we can show that this kind of function is measurable if and not if the set A is measurable ok so we have that then it's a kind of characterization A is measurable if and only if key of A is measurable as a function is measurable ok, let's prove this ok, we start by assuming that key of A is measurable if key of A is measurable ok, we have that we can express the set A has the set for which key of A is equal to 1 so it's a level set ok, this is always holds it doesn't if key is measurable, we have that go inside with the set for which which X in R if you want key of A of X is equal to 1, so this is a level set of a measurable function so this means that that it is measurable, ok this is a level set of a measurable function ok, the covers assume that A is measurable and we want to key of A is measurable so we have to study the set of the type key of A is larger than some alpha ok, so ok, we start by just assuming that the domain adjust a raise, but the domain of key of A was R because this is the check is that the domain is measurable so the domain of key of A is R and R is measurable, ok ok then we fix we pick some alpha in R take some alpha in R and so let us study this kind of set the X in R such that key of A of X is larger than alpha ok, then we have that this set would be actually three, somehow three cases three possibility ok, this would be the empty set if alpha is larger or equal than one ok, because it's never it's never greater than one, ok then it would be then it would be it would coincide with A if alpha is less than one and greater or equal than zero ok, and would be R if in the remaining case, if alpha is larger no, sorry, is less than zero ok, it's negative ok, any case we end up with the measurable set in any case in any case excuse me I mean, if you put here a number which is less than zero this is always it's always true, ok ok, yeah, because I have a the strict inequality ok, in any case we end up in any case means either empty set A or R we end up with with a measurable set ok, now we we study the operation between measurable functions the sum, the product ok, so ok, so let we take two function f and g from A to R ok, for the moment we just assume that the codomain is just R ok, not the extended real line line, so then we will we will study also the general case ok, R measurable function and let's see a constant point in R ok, then we have that also the following the function f plus c c times f f plus g f minus g and f times g R are also measurable ok, so if we perform this operation we still remains in the class of measurable function ok, so we start by f plus c ok, this is f plus c this is f plus c larger than alpha pick some alpha now ok, this is of course is equal to f larger than alpha minus c so, we are done c times f ok, we ok, c times f larger than alpha we can distinguish three cases according to the sign of c ok, for example if c is positive then we have that c times f larger than alpha is equal to f larger than alpha divided by c if c is equal to zero then you have that zero is larger than alpha so, also in this case you have to distinguish some cases, of course the result will depend on how alpha looks like ok, so it would be all a actually, no, you have three and two cases, of course this is all a if alpha is less than zero and will be the empty set if alpha is larger or equal to zero in any case, you yeah yes yes, but f is continuous ah, a zero, ok yeah you can see us in that way yeah ok, just let me finish in this way c is less than zero ok, so you observe that if c is zero the function is constantly equal to zero so it is continuous and we prove before that a continuous function is measurable ok, this is zero less than no, sorry, c times f is ok, larger than alpha, so in this case inequality ok, this case, inequality is reverse so you have that f is less than alpha divided by c ok, and then we have to study the case ok, f plus g ok, the case f plus g ok, we have to work a bit ok, so consider f plus ok, we again we pick some alpha in r we have that f plus g is larger than alpha and call this set b, for instance, ok ok so now since to deal with these cases we will need to to use countable operation between set and somehow to use countable operation we have to use the fact that the rational number are dense in alpha ok, so you have that if, just observe that if f of x plus g of x is larger than alpha ok, then this is trivial f of x actually ok, let me deal with the other set I mean, it's the same it's just for for a sake of convenience, but of course it's the same, so f of x is less than alpha minus g of x ok then, as I told you, we observe that since q is dense in r, we have that this is equivalent then there exists there exists a rational number such that f of x is less than r and less than alpha minus g of x so you have the strict inequality so here you have room to put a rational number ok ok, I call call this inequality r this property r and so what we want to use is the fact that if x belong if x belongs to b then x satisfies satisfies this and also the converse is true ok ok, this is to say that we can characterize this set b ok, this means that that we can characterize the set the set b where f plus g is less than alpha has the countable union over the rational number of these two the countable union of an intersection of x ok, x such that f of x is less than r intersected with the set of the x where f of x no, where g of x g of x is larger no, g of x is minus, so g of x is smaller than alpha minus r ok, so this is these are nice set because this is measurable this is also measurable because we know that f and g are measurable so the intersection are also measurable and now we are dealing just with countable union, so all this stuff will be will be measurable ok, and then ok, f minus g is measurable just comes from combining this step here and the fact that if you multiply for a constant you still obtain a measurable function so just to briefly of that f minus g, you can see it has ok, f plus minus 1 times g no, we are done ok, so what about f times g ok, we we want to express this in we want to express this in this way 1 over 2 of f plus g square minus f square minus g square ok, but we still don't know if this kind of if you take the power 2 of a measurable function is measurable so if we prove this, we are done because then we already prove that f plus g is measurable ok, so consider for instance f square, so what about f square ok, and consider the set f square larger than alpha ok, if you have that if alpha is larger or equal than zero f square larger than alpha can be split as the union of two of this set here larger than the square root of alpha union f of x less this ok, these are two are measurable ok, so the other case if alpha is less than zero this is always true ok, we have that f square larger than alpha is equal to the domain in any case, we know that the domain is measurable ok, so once that we check that if you take the power 2 of a measurable function you still obtain a measurable function then we are sure that while doing this operation we remain within the class of measurable functions so we show that also the product of two measurable functions is a measurable function so I think we span all the cases ok and then we would like to see what can we say about this operation if the co-domain of our measurable function is the full extended real number ok what about if if we have f and g from a to r bar ok so means r union minus infinity and plus infinity ok, we start by this case so when you have s time f ok, somehow this case, the nasty case is when with c equal to 1 and so if you want the interesting case and f is equal to plus and minus infinity ok, in this case there is a convention measure theory in this case in this case to assign this value 0 times plus infinity is equal to 0 which is equal to 0 times minus infinity ok, of course it doesn't always mean in the limit so it doesn't mean that ok, it doesn't mean not mean that if you have if you have a sequence if you have two sequences if you have sequences so there is a cn which tends to 0 and fn which tends to plus infinity in general you cannot say that cn tends to 0 so this is not true so this is just a convention we have precise values c times f ok, what about when you when you have to to consider the sum between two measurable functions in the extended real line ok, so for instance ok, you have plus g so again we use a convention ok, we use we use this convention ok so we have that f plus g is equal to sum constant c fixed when you have the four which are indeterminated, for instance when you end up with this minus infinity plus infinity or the reverse ok, so you put it and c of course when you have plus infinity you put it as plus infinity which is otherwise as I said you put just f plus g ok, otherwise ok, so what about so we can decompose a the domain of this function as the union of two set so somehow there are finite number of possibility for so you have a you see a the domain as a I call it a finite union a infinity ok, so a finite is the finite in this way is the finite union of this set no, actually it's an intersection but is this set is the x in a such that both function are finite ok g of x in r and you can see this has the x in a such that f of x is in r intersection x in a such that g of x belongs to r ok, so why this for instance this is a measurable this is a measurable set because you in turn you can express this has the complement of the set where f of x is equal infinity here I mean plus and minus infinity the complement of the set and we saw that is a measurable set intersection of x in a of g equal to infinity complement, ok, this is measurable as well so the set a finite is measurable what about the set a infinity we will define it in as follows so you have all somehow the other cases f is equal to plus infinity g is equal to plus infinity call this a prime infinity if you want a1 union f is equal to plus infinity g is equal to minus infinity union f is equal to minus infinity g is equal to plus infinity, there are 8 sets so then union f equal to minus infinity and g equal to minus infinity union ok, you have other f equal to plus infinity in r no and maybe you understand how the things are going on actually there are 8 sets then you have f is equal to minus infinity g in r union f in r and g equal to plus infinity union f in r and g equal to minus infinity so there are 8 sets and we call them a2 infinity a3 infinity a4 infinity a5 infinity a6 infinity and a7 infinity ok so when we have to say if our set of the type f plus g larger than alpha whether it's measurable or not we have just to consider a finite a finite number of sets so when you fix alpha in r and you and you study this set f plus g is larger than alpha you can split this in in such a way these are of course x in a x in a finite f plus g larger than alpha union ok union x in a infinity f plus g larger than alpha for this case we can deal as before when we are in the finite case in this case we can deal we can deal as before so when actually we are in this case with r ok we can argue as before more than argue we can directly apply what we proved before ok on the contrary we are here we have just to to compare so we should have something like this our function f plus g takes at most a finite set of values actually we have a in a1 plus infinity is infinity then we use the convention in a2 infinity is equal to c a3 infinity is equal to c again a4 infinity should be equal to minus infinity and then a5 infinity is equal to ok plus infinity then you have a6 infinity which is equal to minus infinity a7 infinity is equal to plus infinity and infinity is equal to minus infinity so basically what I want to tell you is that you have to compare c and alpha just for a finite number of cases ok so for the cases a1 infinity you have that infinity is larger than alpha so this is true and then you continue has a2 infinity you have to say if c is larger than alpha ok and so a that is true so you have all the set or is not true so you have the empty set in any case you have to compare with 8 cases and you have so we have a finite number of cases and you always end up when you have to say where this is the finite you end up with with a measurable set ok yeah sure please go and ok so you can do the same let me repeat this argument because it's very long it's nothing too deep but for f times g ok no no you can go I will continue for the minutes but you can go ok when you consider f g r and you have to deal with the product I mean you can argue similarly so you have 0 if use the convention if f times g is of the form for instance 0 times plus minus infinity and then will be plus minus infinity if f and g are equal to plus or minus infinity and then ok this is the easy case f plus f times g if f is less than infinity and g is less than infinity ok ok and also in this case you can define a finite as the case when this case and you can define ok you can see a the union a finite union a0 infinity and this a0 infinity somehow collect this first case just start to write some of this but then you understand how it is f is equal to plus infinity and g is equal to plus infinity union f is equal to plus infinity and g is minus infinity then you have the reverse and then union you have the cases in which f is equal to 0 and g is equal to plus infinity union ok f is equal to 0 g is equal to minus infinity and you continue as you have again 8 cases so when you have to compare alpha the test the test values you have to compare alpha just with a finite with a finite numbers of values and you always end up with a measurable set ok so you argue the argument is completely analogous with what we done for for the sum ok it's very long I don't want to to repeat it again ok so now we have some minutes left who of you has to go to do some of you has to ah no ok may I raise here ok so remark so do you think that there exist no measurable function how can you construct a function which is not measurable you use the no measurable set so you have to use I mean the way is to use the no measurable set that we construct ok but then how could you define has we saw that a characterization about characteristic function of set you have to combine the fact that there is a set T which is not measurable T21 or rather you can use it as the characteristic function of the no measurable set ok ok this is ok this is not measurable ok this is an example of the function ok what can we in the last minutes we can prove this proposition ok so let S A to R measurable then S measurable only if you have that for any B to M sorry pre-image of infinity pre-image of minus infinity measurable excuse me ah sorry ah function ok ok so ok I think we here then to to require A fact that A is in M ok actually somehow this is related with with the for instance she your name Torejla before she suggest this example has a no measurable function the constant function define on T which is completely correct here the fact that for instance F is not measurable is also because the domain is not measurable but this is correct so these are two functions this is why we when we want just to check the second part so the part concerning the the level set if you want we have still to to require this ok so let's see this implication so we assume that F is measurable ok we saw that if B is we start by B open set we saw that we can express this as the countable union of this joint open set of the type type A i B i ok these are this joint open interval sorry open interval ok so if you look at the preimage of B this is the union S minus 1 of A i B i and then we have that because the preimage preserve preimage of this joint set preserve the countable union ok the preimage preserve what is that F minus I know yes yes sorry it's the preimage I'm sorry preserve countable ok so we can understand how it's going on you have that this preimage can be seen as the intersection of of these two sets and so you have that the preimage of B is ok it's the countable union of ok of this set S minus B i and everything is measurable ok and then ok what about the preimage of this plus infinity this is the intersection F is larger than N and again everything is measurable ok ok now we deal with the vice versa ok we want to prove that F is measurable if these two are satisfied so we so we check, we pick some alpha enough and we check if this set is measurable so but how we can express this set we can express it as the preimage of alpha plus infinity union the preimage of plus infinity ok ok but we we assume that these are both measurable and so we are sure that also this set here is measurable and so we this concludes this concludes the proof so i think that for today we can stop here


It is time for This Is My Bolo. Stay tuned for details on how you can participate in the next video. Everyone is invited to share a Bolo and shout out their YouTube channel or their eBay store. And again, more details to follow. I sold this puzzle for, Hey Bolo Buddies, thanks for watching. Here's for online, little pony. Amazing, right? All right, let's get started. It is time for This Is My Bolo. This is one of my favorite videos. It's where I invite my featured members to send me a 60-second video of a Bolo they sold, an item to be on the lookout for, an item that they picked up cheap and sold for a great profit. This can be a big money Bolo or an unexpected Bolo, maybe something we wouldn't expect to be looking for. So I'm gonna start by showing you mine. And I sold this puzzle for $300. And it is a Springbok puzzle. And it originally was $7. It's considered a king size. And it's a thousand pieces. The buyer was all in for $332.37 with tax and shipping. I got that offer in less than a week. So this sold really, really fast. I picked this up at an estate clean-out. I purchased a whole car full of stuff for one price. So my cost of goods for this was very, very low. It was definitely a nice surprise. I had no idea it was gonna be worse so much, but anytime you can find big puzzles, like this is a thousand pieces, and vintage, still sealed. It does have some areas where the seal is broken. I just disclosed that in the listing. And this is definitely a big money Bolo. This is called the Christmas House. I am going to share with you some more Bolo's, but I'm not actually doing the sharing.  Okay, I'm gonna explain how this works.  Oh my goodness, wait until you see these. Okay, so if you would like to participate in the next This Is My Bolo, I need your video by the end of August. The end of this month, guys.  You're gonna show me the Bolo, either hold it up or pop up a screenshot, and you're gonna tell me where you got it, what you paid for it, and what it sold for. And make sure if you have a YouTube channel that you shout that out, or if you don't have a YouTube channel you are still welcome to participate, you can shout out your eBay store. So I'm excited to see all your Bolo's. You can send those to me on Instagram Messenger or Facebook Messenger. And again, get those in by the end of August. Bolo's you're about to see are from my featured members. They will all be linked down in the description with their YouTube channels and their eBay stores. Definitely go and sub them up. They are big money Bolo finders with tons of awesome reseller content. Let's get started. Hey guys, I'm Kristen. I'm a Roll Squirrel. You can find me on YouTube and Instagram and Facebook and whatnot and everywhere as a Roll Squirrel. The next Bolo that I wanna share with you blows my mind. These, this is a pair of vintage undies. Little sheer panties that are made in the US and they're union made. But these undies I sold for $89.99. That is a full price sale and they were listed less than an hour.  So be on the lookout for vintage undies. Thanks Courtney for putting this together. I can't wait to see what other Bolo's everybody else has got. Hi, my name is Terry. I go by the name Mindful Recycle on my YouTube channel and in my eBay store. Today's Bolo is this very large Atlas. I purchased five of these from an estate sale this past weekend and I paid $5 each for them. Ran the console while I was there and was shocked. I listed my first one for $475 and it sold within 24 hours. I listed my second one for $599. It also sold within 24 hours. Getting ready to list my third one we'll see what happens. So keep your eyes open for the books. It seems like whenever I go to estate sales people aren't going for the books and I always seem to find treasures over in the books. So go find your Bolo. Thank you, Courtney. Hey y'all, it's Rhonda from Got Junkin' Our Trunk and today I have a Bolo for you. We know as resellers that we always want to look for rushed and especially if we've dealt in plush very much. We usually see like Santa Clauses. There's some of the different type of animals that we'll find. Well, today I have one that you probably haven't found at least in this condition. I found Daisy the cow. She's kind of a little bit harder to find but in this condition you usually don't see her this moth eating. I bought her at a yard sale. I ended up paying $2 for this thing and here's what she sold for. Y'all always want to look for the rushed and that's your Bolo for today. Happy Junkin'. Hello, my name is Cindy. I'm the owner of the Warepix Fly Farm out in Lynn, Missouri.  Things that we purchase at auctions. I purchase storage units and then some things we have donated. My Bolo this month was very unexpected. A lot of times when I get things in I'll throw the smalls into a box or in the baggie, set them to the side, try to list my bigger things first, get them out of the way. Anyway, I ran across this, a little poker chip. Now I had seen on other Bolo videos where poker chips could go for pretty decent money. I looked this one up, couldn't find it on eBay, found it on Worst Point. The last one sold for over $300. They're mine up for auction and boom, it sold for over $400 to a collector. So keep your eyes open for these poker chips. Hey everybody, this is Matt. I go by MSG's Treasures, both on eBay and YouTube. I just want to thank Courtney for doing these videos. They're really cool. I have a monster of a sale to show you. Here we go. After struggling to find a good Bolo for this video, all these sold for me over the weekend. I haven't had any home run sales in quite a few weeks. And here we go. The planes are all Franklin Mint. You see prices on the screen and these two tanks here are King and Country and Fogarty miniatures. Fogarty and King and Country is a pretty rare Bolo. I've only found those the one time in four years of doing this full time. They've even heard of the brand. Franklin Mint is more easier to find. Any die cast with Franklin Mint, if you can get it cheap enough, buy them. I'm Danielle here at Yard Sales and Dreams and this is my Bolo. This item is actually getting shipped out. I sold this sweater for $90 and I picked it up for $1. How much are your sweaters? A lot of dollar on that one. All right, that's fine. I bought that for an ugly Christmas sweater. Did you? All right, you guys. So this sweater sold for $90 and if you guys are not familiar with this kind of sweater, this is a kooky lookalike. It's called a Bouchie, I believe, but there's very many brands out there that make a sweater that look like a kooky or koogee. However you wanna pronounce it. These lookalikes, the fakes do sell anyway, so you just have to disclose that it has a different tag and this one, I believe I said it already, Bouchie, B-A-U-C-C-I, I think, something like that, but anyway, it's all for $90, so, and I paid a dollar. Hey guys, my name is Tanya and I'm just a girl out here making a living thriftin and today's Bolo is an empty box. That's right, this G.I. Joe empty box. I went to an estate sale, I picked this up, it had a price tag of $30 on there. It did come with a G.I. Joe inside, however, the G.I. Joe is not the Marine G.I. Joe, so I split the two up and I ended up listing this box all by itself for $137 with free shipping. The box is not perfect. It actually has several flaws to it. Some of the glue is bad. Some of the tape has come off right here, which I made sure to show all of that in my listing and I still ended up selling this within 30 minutes of posting it. It is a 1964 G.I. Joe Marine. Hi Courtney, hi everybody that's watching. I'm Yvonne, Mom Knows Treasures. My Bolo that I wanted to share was a Made in Italy duvet cover. I sold it for $225. And if you haven't started yet, I would highly recommend selling linens. I've sold quilts, I've sold vintage sheets, I've sold sheet sets, blankets, San Marcos blankets, all these things sell. People will buy them new and used. This one was sold for $225, but they run the gamut. If you can find linens at a good price, pick them up. Okay, thanks everybody and good luck. Out there sourcing, bye. Hey everyone, Stuart with Franklin Hill Ventures and this is my Bolo. It is this Derue Leatherinostrus from Germany made in the 1960s and if you haven't heard of it, I don't feel bad, I hadn't either. I was going through some stuff from an estate that I bought out probably two or three years ago, had a few boxes left over my garage and that was gonna either take stuff to the dump or to the thrift shop, happened to see a name on this, looked it up and boy was I surprised. I listed it on eBay and within a couple of hours, it sold for the full asking price of $110. So if you have stuff in your garage that you haven't looked up, make sure you're looking up comps. I don't just assume that something's not worth anything based on the way it looks because I certainly didn't think this hollow piece of leather would be worth $110, but it was. So hope you enjoyed and everyone have a great day. I wanna introduce myself, my name is Abigail. I have a YouTube channel. It is Abigail's Artful Abode and I own a vintage and antique store. I have two antique booths in Seattle, Washington and I sell on almost all of the major platform. Today, I wanna share my Bolo. I don't know a thing about cameras but they're very easy to research. I picked this up. It was these lenses. I think there was four of them. They were in a box. I paid a dollar for the whole box. So I'm maybe into a like 25 cents piece. We listed them for $50 a piece. Within a couple of days, we got an offer from one buyer who wanted all four and we took the offer of $160. So I have to say that was a great Bolo. Sometimes cameras and camera accessories are bread and butter. Sometimes they are Bolo's. Hey guys, this is Siever from Late Threads and this is my Bolo. It is an Ermin needlepoint tapestry, Noah's Ark cushion. I picked up three of these and the state sale for $20. It sold all three of them within just a couple hours and they sold for $59.99 and $69.99. They are artist design, huge Bolo. If you haven't seen these before, definitely worth looking up and thank you, Courtney. Hi again. My name is Rebel and I'm the Rebel reseller and today I am doing This Is My Bolo. I start all of my listings on eBay but I do cross list to several platforms. Etsy is being won and this item sold on Etsy for me. This is a Sanrio Peckle Duck nylon plush 13 inch from 1995 and it sold for $47.45. Sanrio is a brand that you really should be looking out for but especially when it's just not Hello Kitty, it's other characters and I actually had to use Google Lens to figure out who this was but also plush that is made of this nylon or parachute material, definitely check your comps. Again, I wanna thank Courtney for all that she does for those of us who are trying to make YouTube videos. I really appreciate everything you've done for me. Thanks, bye. Hi guys, Farm Girl Scavenger here. It is hotter than the blazes here in Southwest Washington. We are at 96 degrees today and I wanted to show you something that I think you should be on the lookout for also know that because of the fact that most people price these at $19.95 or less, you're gonna have to sit on it for a little while but I want you to understand you need to know the worth of your item and you need to price it at what it's worth and sit on it. Otherwise, you're bringing the market down not only on a collectible but you're bringing the market down on something that there aren't a lot of out there. So I purchased this in January in LA when my grandson and I went down to LA to visit his dad. I went to the LA Flea Market with Jenny from Spruce and Gussied and I saw this across the way and I paid $10 for it. It is a vintage 1950s, Gilner, California pottery, Pixieware, happy people boot planter and they're hard to find. And you'll see a lot of people with this, these Pixie, these Gilner Pixie items are priced at $19.95, $20.95, $9.95 and it just blows my mind. I priced this high at $129.99. I got a lot of giggles and grief from people and I held on. Now, did I think I was gonna sell it at $129? I don't know, you never know. There may be that one person. I got an offer last night for $75 and I took it. This is shipping out today for $104.50 all in. I hope that helps you and know your worth, know your items worth and let's quit bringing down the market on collectibles. Thanks, hope you have a great day. Bye. Hi, it's Rhonda from Got Junkin' Our Trunk and I have a bolo for you today. When I was a kid, we used to have door-to-door salesmen that would come around selling health and beauty aids, vacuum cleaners and such and my mom typically didn't buy a whole lot of things when people would come to the door. But when the Stanley person would come by, she would usually get some of their products for hair and things like that. But when I was about, probably about six or seven years old, maybe eight, we needed new brushes and so mom bought my sister and I'm brushes and this was my brush when I was a little kid. It is a Stanley brush. As you can see, this is like a plastic acrylic type handle nylon here and of course, it's been used and all of that. This actually is a bolo. It's kind of like the booty brand. If you look for the Stanley, you can usually sell their products pretty well too. Here's what this sold for. So keep your eye out for things like this. Don't pass them by. Happy junking. Thank you all so much for being here. I am going to include links to all of these channels, all of their eBay stores down in the description of the video. Please take a minute to go and subscribe to these channels and check out their eBay stores. They are all amazing bolo finders. Thank you to everyone that participated and again, I'm going to open this up to anyone that would like to participate. Please just send me a 60 second or less video similar to the ones you saw in this video. You tell me where you got the item, what you paid for it and what it sold for and I will feature it in a video. So if you can get those to me over on Facebook Messenger or Instagram Messenger by the end of August, I will get those videos put out in September. Thanks so much for being here. Thank you for watching. Be sure to also subscribe to my channels. I have three of them, Bolo Buddies, which is where you are now and I also have sourcing with Bolo Buddies and I also have reseller testing Bolo products. I would love for you to subscribe to all. They are all linked down below in the description. Thanks for watching.