That concludes general questions. Before we move on to First Minister's questions, may I invite members to join me in welcoming to the gallery Dr Gabriel Andretta, President of the State Parliament of Lower Saxony. The next item of business is First Minister's questions. I intend taking both constituency and general supplementaries after question 2. Members wishing to ask supplementaries please press the request to speak buttons during question 2. I'll keep a note of members who press and take further supplementaries if we have any time in hand after question 7. Members wishing to ask a supplementary on questions 3 to 7, please press during the relevant question. I call Douglas Ross. Thank you very much, Presiding Officer. After months of worry about what would happen, last night we heard that the train strikes have thankfully been averted. While it's a welcome relief for commuters across Scotland, it should never have taken the 11th hour to secure. So can the First Minister explain why the additional funding and urgency was only found to resolve this now, as world leaders are coming to town for COP26? And why are they more important than ordinary Scots who have had to put up with this since March? First Minister. Douglas Ross clearly hasn't looked at or understood the nature and the detail of what was agreed last night, but I'll come on to that in a second. A very reasonable offer was made to the rail unions. Negotiations have been on-going for some time. I think that the reasonableness of the offer is evidenced by the fact that of the four rail unions that were party to these negotiations, three of them had already accepted that offer. The outstanding one was the RMT, and agreement was reached, I'm delighted to say, last night. The basis of that agreement is a one-year deal. This is where I don't think that Douglas Ross has looked at the detail. There is no additional funding in that one-year deal. The deal that was offered for the first year is the same as the one that was offered to the RMT earlier this week. The difference is for the second year, which the other unions have accepted. There will be further negotiations to come. I think that this is a good outcome for the travelling public across Scotland. It doesn't simply remove the prospect of a rail strike over the period of COP. It also resolves the Sunday strikes that have been on-going for some time. It's a good outcome and one that I'm delighted to see secured over the course of the last 24 hours. Douglas Ross. Does the First Minister really expect us to believe that there was no funding involved in that? Suddenly, the rail unions just suddenly decided to accept it. I was very clear in my question, and I hope that the First Minister might have taken the opportunity to apologise to people across Scotland who have faced this disruption since March. They have been waiting for a resolution for months, and only now, as world leaders are about to come to Glasgow, has this been resolved. COP26 presents a huge opportunity to tackle climate change, but for working people across Glasgow it will disrupt their daily lives, even with this RMT strike cancelled. The list of road closures is considerable, and there is massive potential for traffic to grind to halt. Can I ask the First Minister if she is confident that Glasgow City Council and the Scottish Government have done everything they can to minimise the disruption to commuters and local residents in Glasgow trying to get on with their daily lives? First Minister. Firstly, before I come on to the questions relating to COP more substantively, let me finish and conclude on the points relating to rail. In passing, it's worth just pointing out that, as far as I understand it, and I'll be corrected if I'm wrong here, where Douglas Ross's party is in power south of the border, rail workers are getting no pay increase anywhere near the pay increase that has been offered to rail workers in Scotland. So not for the first time there is something of an irony. Secondly, and this is where I really would suggest that I would have thought that he would have done it before coming to ask these questions today, but given that he clearly didn't, he might want to do it after he's asked these questions. The agreement that was accepted by the RMT last night is virtually identical to the deal that had already been accepted for the other unions for that one year. That is the position. It is the same deal that had been on the table at the weekend for the RMT. I am delighted that that agreement has been reached because it now does remove the prospect of strikes. Moving on to the issue. Firstly, I think that COP, I hope that COP is successful in terms of the objective of making real progress towards tackling climate change. All of us should want to see that success over the next two weeks. The Scottish Government, working with the UK Government, with the United Nations, with Glasgow City Council have put in place appropriate contingency measures to ensure the successful logistical operation of COP. We will not be complacent. Our resilience arrangements are stood up. There will be day-to-day monitoring of all the different aspects of this, but I am as confident as it is possible to be that those arrangements are appropriate. Of course it is not a Scottish Government event. It is a United Nations event. The UK Government is the formal host and we are working closely with them. To the people of Glasgow, and I am a resident of the city of Glasgow, I am a representative of part of the city of Glasgow, there will be disruption and inconvenience over the next two weeks and I know that that will be regrettable in many ways. I think that the majority of people in Glasgow understand the importance of the COP 26 summit for the future of the planet. I want to take the opportunity of wishing the United Nations negotiators, the UK COP presidency and everybody attending in that official capacity every success in reaching a deal that does put the future of the planet and generations to come first. Let's just start with what the First Minister said in that answer, and crucially what she didn't say. I hate to break it to the First Minister, but you can't be virtually identical. Identical means they are the same, so if they are virtually identical, then they are not the same. There was clearly a difference for that to be accepted, but what the First Minister didn't say in her first answer or her second answer was sorry. Sorry to the people of Scotland who have been waiting for months for her Government to step up and resolve these issues. It has been resolved now, but it should have been resolved far sooner because people have been struggling with these strikes since March. The First Minister did accept that there will be disruption and inconvenience for people in Glasgow, and there is that real potential for disruption from protest groups at COP26 that risks public safety. Just this morning, Extinction Rebellion said that they are planning deliberate disruption with the most impact possible. We all respect the right of protesters to express their views, but we cannot sit back and allow deliberate and dangerous disruption of people's lives. Will the First Minister reassure people across Glasgow that there will be a zero-tolerance approach to protests that disrupts people going to their work, including doctors and nurses, and ambulances carrying people in urgent need of medical care? First on rail, what Douglas Ross is really displaying here, if truth be told, is a real disappointment because he would rather have seen it continued. He wanted to know the detail again, given that he did not check the detail before coming into the chamber today. The deals already agreed by the other unions consisted of, for the first year, 2.5 per cent backdated for 2021, a £300 COP26 payment and a rest day working agreement. The deal agreed with the RMT last night, consists of 2.5 per cent backdated for 2021, a £300 payment for COP26 and a rest day working agreement. That seems to be pretty identical to the one that had already been agreed by three of the four unions. It is good news for those who travel on our railways. It is good news for the Scottish population, which is probably why Douglas Ross is so deeply irritated by it. Obviously, it is for Police Scotland to decide the appropriate approach to the policing of demonstrations. The chief constable, who I will be having further discussions with over the course of today and tomorrow, has been very clear that there will be a sensitive policing operation that will do everything possible to facilitate appropriate and peaceful protest. However, Police Scotland will respond to any protest that seeks to break the law and, of course, disrupt people beyond what would be considered reasonable. People do want to come and make their voices heard. I think that that is understandable, given the importance of the issues under discussion. However, I would say to people looking to come to protest in Glasgow, do it peacefully and do it with the recognition that the people of Glasgow are agreeing to host this conference, are suffering some disruption because of that, so do not add to that disruption for them. However, let us all get behind those who will be negotiating. I hope a good outcome for the future of the planet. Douglas Ross speaks about disappointment and irritation. If she wants to look for disappointment and irritation, it is commuters who have been putting up with this since March. We have now had three attempts for the First Minister to have some humility and accept that the problems of those strikes have affected people up and down Scotland for months. Does she take the opportunity to recognise the disruption that has caused to people across Scotland and the fact that it has been resolved at the last minute? It looks like it is more important for COP26 to be suitably sorted in terms of travel arrangements and not for people across Scotland. She also mentions the protests that are expected over the next couple of weeks in Glasgow. There have been suggestions from some public figures, including one of Nicola Sturgeon's own ministers, that some unlawful protest will be tolerated. It is one thing to be frustrated by the lack of action on climate change, but it is another thing entirely to take that frustration and use it to disrupt people's lives. We all want COP26 to be a success. It is not just an opportunity to tackle climate change, it is a once-in-a-generation chance to highlight the best of Glasgow to the rest of the world. It has already been a rocky road getting to this point from strike threats to hospital appointments being cancelled to the well-known problems within collections and concerns over wider travel disruption. Is the First Minister now fully confident that Glasgow is ready to grasp this opportunity? Yes, I am. Obviously, the UK Government has a big part to play here. Douglas Ross is sounding a bit disappointed that the UK Government decided to bring COP26 to Glasgow. Perhaps he wants to direct some of these concerns to them. Those are serious issues that the Scottish Government has been focused on with our partners, which, in the case of COP, includes the UK Government, Glasgow City Council, the United Nations. I met just yesterday with the UN lead negotiator in COP26 to discuss, obviously, some of the logistical issues around it but also the substance of the negotiations. On the issue of rail disruption, the Scottish Government has been supporting ScotRail to bring an end to any disruption. I always regret disruption that is caused by disputes of this nature, but I do think that the offer that has been made to rail unions was a reasonable one, evidenced by the fact that three out of the four unions had already accepted it. I am glad to say that we reached agreement with the fourth of those four unions last night to take away the prospect of strike and to end the Sunday disruption that has been suffered now for some time. Finally, on the issue of protests, in a constitutional democracy it is not for politicians to decide how the police demonstrations it is for the police to decide how they appropriately police demonstrations. If what Douglas Ross is asking me is, do I have confidence in the ability of Police Scotland to do that appropriately and sensitively and with the interests of the people of Glasgow in Scotland at heart, then yes, I do have confidence in Police Scotland to do so. Thank you. Next week, the eyes of the world will be on Glasgow as leaders gather for our last great chance to avert the climate emergency. It is in all of our interests and those of future generations that COP26 succeeds. We have the opportunity to strike a historic Glasgow agreement, but the proposals currently on the table currently would still lead to over two degrees of global warming. That simply isn't good enough. That's why it's important that political leaders both at home and abroad turn their words into meaningful action. Does the First Minister agree that that means leading by example? Yes, I do agree that. Scotland does lead by example. That is not to say that we have not got more work to do because we absolutely most definitely have. Our own statutory climate change targets are consistent, more than consistent indeed with the Paris agreement. Therefore, that gives us the ability to apply pressure to others. We are not, unfortunately, directly at the negotiating table, but we have a considerable degree of influence, not least through our co-convenorship of the Under 2 Coalition, bringing city, state, devolved Governments together to maximise maximum pressure on those discussions. We must lead by example. There is a big job of work to do to keep 1.5 degrees alive, which is the aim of the COP26 summit. There is a gap on emissions. Right now, there is a gap on climate finance, and I know that the negotiators are very focused on trying to close that gap as much as possible, and that's what we've got to hope emerges over the period of this summit. I'll come back to the national record, but let's talk about what it means locally. First Minister, I love Glasgow. It's my home. But frankly, it's been let down by the SNP, who can't even get the basics right. Tons of waste piling up on our streets, fly-tipping on the rise, and over a million rats. Glasgow deserves better. So while Nicola Sturgeon lectures the world about the global environment, she is turning a blind eye on the environment that glass regions are living in every single day. Tomorrow I'll be joining cleansing workers who have been on the front line throughout this pandemic. They have been crying out for months for Nicola Sturgeon to tackle the waste crisis, but they have been repeatedly ignored. So will she join me tomorrow in Glasgow to hear directly from them about the challenges they face every day? First Minister. I'll be working hard to make sure that the Scottish Government is doing everything to support cosla and local authorities' hope to reach an agreement with trade unions to resolve the issues here. That's my job and responsibility. I don't shy away from the problems and the challenges that cities like Glasgow, because Glasgow is not unique here, faces in the times that we live through right now. I stand here and allow Glasgow, one of the greatest cities in the world, to be talked down for political purposes in the way that Anna Sarwar has disgracefully been doing in recent times. Anna Sarwar. We're going to be talking next week about making history. That was probably a historically out-of-touch answer from the First Minister. She is basically saying to glass regions, that's just as good as it gets, and that's frankly not as good as it gets. She talks about talking to people that are world leaders. These are people on the front line leading the fight against the environmental crisis, and she should be taking them much more seriously. Because she needs to lead by example. Nicola Sturgeon is right to say that we need credible action, but while she talks about the need for more public transport and to get people out of the cars, her Government is cutting hundreds of train services. She lectures the world on the global environment while cutting cleansing budgets and neglecting the local environment. Her SNP Government has missed its renewable heating target, missed its gas emissions target, three years running, and she promised 130,000 green jobs by 2020, but we have just over 21,000. We all want COP26 to be the moment the world comes together to stop the climate catastrophe. So when will Nicola Sturgeon stop talking about the credible action and start delivering it? First Minister. Let's take each of these in turn renewable heat, where we are seeing an increase in renewable heat. The decline reported yesterday driven by the reduced output from large biomass systems as the energy saving trust noted yesterday actually massed growth in renewable heat output from other technologies, including and in particular from heat pumps. Take greenhouse gas emissions in Scotland. We have decarbonised as a country in recent years faster than any G20 country. We've reduced emissions by 51.5 per cent. Yes, our target said that should have been 55 per cent, which is why we're publishing a catch-up plan, which the law requires us to do. We've decarbonised greater and faster than most other countries in the world. That's why we lead by example. In terms of rail services, there has been a consultation. Scotland Rail are now looking, of course, at all the responses to that consultation to make sure that we've got rail services fit for the future. It's this Government, I'm proud to say, because Labour, when they were in government, didn't even allow us to have the powers to do it, that is going to re-nationalise Scotland's railways to make them fit for the future. Thank you. We'll now take supplementary questions, and I call Jim Fairlie. Thank you, Presiding Officer. In 2014, we were told that the people in Scotland benefit from the UK's influence on the world stage. Yet under Westminster's control, decisions are constantly taken that affect our lives, but without any serious consideration to the interests of the Scottish people. The latest example is the New Zealand trade deal. Can I ask the First Minister for her assessment of the possible effects that this agreement will have on our farming industry and on the wider Scottish economy? The proposed trade agreement with New Zealand represents a significant opening up of our agriculture market to imports of New Zealand agri-food, which is produced to lower costs and, crucially, will do nothing to offset the damage to our economy caused by Brexit. The UK Government's own economic analysis concluded that a UK-New Zealand trade agreement would have a 0 per cent impact on UK GDP overall, while the deal would deliver a contraction of 0.5 per cent to GDP in Scotland's agriculture and semi-processed food sector. We were not involved in those negotiations, but this proposed deal is evidence that, when it comes to negotiating those trade deals, Scotland's interests are nowhere on the radar of the UK Government, and it is utterly disgraceful and yet another downside implication of the Brexit disaster. I call Douglas Lumsden. Thank you, Presiding Officer. As the First Minister knows, I along with many other Scots are taking part in the Novavax vaccine trial. It's now over two weeks since trialists in other part of the United Kingdom have been contacted and offered an alternative vaccine, but in Scotland there has been silence. Can I ask the First Minister whether the Scottish Government will follow the UK Government in offering trial participants two doses of an alternative vaccine? That would give them clarity and peace of mind that they are appropriately protected. First Minister, discussions are on-going on that matter. We have absolutely made clear that there will be no disadvantage to those who took part in vaccine trials. We are deeply grateful to those who did so. I know that the health secretary is having discussions with the Secretary of State and the chief medical officer later today and, of course, will update Parliament as soon as possible. Jackie Baillie. A 94-year-old woman in my constituency had to cancel her booster vaccination appointment because she caught Covid. Thankfully, she is now recovered, but the family have been trying to re-book online and on the telephone with the NHS informed booking service. They tried over three days, phoning several times a day. The operator told her yesterday that the system was down. It has been down all week and they do not know when it will be fixed. When the First Minister was telling everybody to phone NHS informed this week to get an appointment, the system was not functioning. Can she tell us when it will be fixed? First Minister. It is resolved, as I understand it now. There was a problem with it if Jackie Baillie wants to send the health secretary details over a particular constituent. We can be in touch to make sure that they have what they need to re-book the appointment. There will be issues like that from time to time. They are regrettable. We fix them as quickly as possible, but as of yesterday, almost 600,000 booster vaccinations had happened across Scotland. Every day, there are thousands of booster vaccinations taking place, and that is to the great credit of everybody administering the scheme across the country. Paul MacLennan. The First Minister will be aware that the people of Afghanistan are currently suffering through a humanitarian crisis with more than half of its population facing acute hunger as the country is gripped in one of the world's largest food shortages. The UN has warned that a harsh winter looms with over 23 million Afghans will go hungry as a result of conflict and economic downturn. Will the First Minister express her solidarity with the plight of the people of Afghanistan and call on the UK Government to work quickly with the UN to come to the red? First Minister. Yes indeed, I am sure all of us would want to express once again our solidarity with the people of Afghanistan. The Scottish Government has announced humanitarian funding through our own humanitarian emergency fund. We did that just last month. We are also welcoming people from Afghanistan and helping to give them refuge here. We all want to do everything possible to help, and the Scottish Government is absolutely focused in making sure that we do that. Liam Kerr. Aberdeen City Council is £6 million out of pocket as the Scottish Government has not paid vital Covid grants from over a year ago. Ministers have pushed back the date for payment on three separate occasions. When will Aberdeen Council get the money due? Councils are getting every penny that they are due. I will look into the particular issue of the timing of payments that Aberdeen City Council in right to the member, but councils have had tens of millions of pounds of Covid funding and they get, as I say, every penny they are entitled to. Paul O'Kane. Last week, one of my constituents, Miss Cooper from Bishop Briggs, went to get her Covid-19 booster vaccine. She arrived on time for her appointment, but the vaccination centre had no available disabled parking, no managed queuing and no seating for waiting patients. Miss Cooper, who is 83, has dementia, diabetes and reduced mobility and was made to wait for an hour and a half outside in the cold and rain. Miss Cooper's daughter now tells me that her mother is afraid to go for any future vaccination. With the elderly forced to wait in such conditions, it is turning people off from getting their vaccination at a time when it is more needed than ever. What is the First Minister doing to ensure that our elderly and most vulnerable citizens are vaccinated quickly, safely and that nobody's mother or father has to wait for hours in inclement weather, which I am sure she will agree is unacceptable? First Minister. I know that the health secretary has raised with NHS board chairs the issues around people having to queue for appointments and it is really important that that does not happen. Obviously, I would accept that what has been recounted there is not acceptable for any elderly person, but I do think that it is also important to recognise the huge success of the vaccination programme, including the booster campaign that is under way right now, literally as we speak. There are thousands and thousands and thousands of people being vaccinated with booster jags and that is a good thing because it is a vital part of our protection over this winter period. Health boards are working hard on that. Vaccinators are working hard on that and the Scottish Government is doing everything and will continue to do everything to support that programme. Thank you, Presiding Officer. First Minister, NHS boarders are currently sending out booster vaccination invitations. Can the First Minister confirm to my constituents that if you register with a boarder's GP that even if you've had one or both vaccinations in England, for example, you will be notified of this appointment by NHS boarders? Yes, that is the case. The appointment notification will be through the NHS in the boarders. Obviously, anybody who doesn't receive their notification when they believe they should have done the processes that are there allow them to check what and if there is a problem with that. Pam Gossel. Sorry, Presiding Officer. A recent report revealed that 53 per cent of university staff showed portable signs of depression and that one in five academics were working at least two extra days per week. What action is the Scottish Government taking to help universities and colleges to reduce the workload of staff as restrictions persist? First Minister. We recognise that there has been a mental health impact on many, many people in different sectors across the country. We are taking steps in a range of ways to improve mental health provision. First and foremost, it is for universities and colleges to make sure that they are looking after the wellbeing of their own staff. Through the discussions that we have with the sector about funding and other support, we will, as we will do in all sectors, make sure that mental health issues are properly catered for. To ask the First Minister in light of COP26 what action the Scottish Government is taking to showcase the role that Scottish agriculture can play in reducing carbon emissions. First Minister, I am sure that Rachael Hamilton will join me in welcoming the announcement this morning by the rural affairs secretary of a national test programme of measures to kickstart action by farmers and crofters to reduce their carbon emissions. The rural affairs secretary also announced yesterday that the Agri environment climate scheme will reopen for funding in 2022 to support the ambition of doubling the amount of land under organic management. We have also published a consultation building on key themes from the farmer-led groups that will help to inform future work. I also hope that Rachael Hamilton will have noticed what we are not doing here in Scotland. We won't be selling out Scotland's farmers and food producers in trade agreements which threaten to undermine standards and prices. We won't be cutting the level of payments that our farmers receive and we will fight UK Tory plans to treat agriculture payments as subsidies unlike any other jurisdiction in the world which further undermines the potential support available for farmers and crofters in Scotland for producing food. Rachael Hamilton Presiding Officer, Rishi Sunak's budget yesterday delivers for Scottish farmers and crofters providing an additional £1.9 billion which will be beneficial in fighting climate change. We also welcome the Scottish Government's commitment for further axe funding, a long-standing Conservative demand, but farmers require urgency on the detail of that extended funding. Scottish agriculture has some of the most environmentally friendly practices in the world yet there are concerns from livestock producers that the industry will be used as a scapegoat. Has the Scottish Government ever considered allocating funding to accelerate the reduction of cattle numbers yes or no, First Minister? First Minister. I think that our agriculture sector has been used as a scapegoat, but not by this Government it's been used as a scapegoat by the UK Government and frankly betrayed and let down by the UK Government. We know that agriculture is an emitter of green house gases. It must play its role in us meeting our net zero target, but we will do that in a way that supports the sector appropriately to do that and we will not short change the sector in the way that the UK Government has. Thank you. Question number four, Kenneth Gibson. Thank you, Presiding Officer. To ask the First Minister what the Scottish Government's response is to that. There are aspects of the UK Government of course that we welcome but there is also much to be disappointed in the UK Government. It does not do nearly enough to address the cost of living crisis that many individuals and families across the country are facing and it will leave the Scottish Government with less resources in every year of the spending review than we have at our disposal in this year. Although it results in considerable challenges for the Scottish Government but of course we will set out our own budget plans shortly. Kenneth Gibson. I thank the First Minister for that answer. Millions face a squeeze on living standards over the coming year, where the chancellor cut taxes on bank profits he failed to introduce measures to help households already struggling with rising prices on food and fuel. The tax burden is now at its greatest since the 1950s with national income. Does the First Minister agree with Paul Johnson, director of Institute of Fiscal Studies, that this is actually awful? Yet more years of real incomes are barely growing, high inflation, rising taxes, poor growth, keeping living standards virtually stagnant for another half decade, anti-poverty charity, Z2K, that there is absolutely nothing in this budget for the 3 million plus whose disability illness are caring responsibilities mean they can't work and will be hard this winter? Does she further agree that Mr Gibson, could you please hold on for one minute please? Sorry. That was an exceptionally long question, but I will decide whether or not we will hear your exceptionally long question. I'd be grateful if the First Minister has heard enough to respond to that. I would ask members to bear in mind that there is a great deal of interest in this session. I'd like to get in as many members as possible. First Minister. I suspect I know why the Tories didn't want to hear Kenny Gibson's question because there were some deeply uncomfortable truths in it for them. When I gave my initial answer to Mr Gibson, there was laughter from the benches of the Conservatives when I said that we would have less money in every year of the spending review than we do have in this year. Let me just give the detail of that between this year and next year, Scotland's resource budget is being cut by 7.1% in real terms. The equivalent reduction for our capital budget is 9.7% in real terms. That's the reality and I'm not surprised that the Tories don't like it. First Minister, if you could just give me one second, I would very much like to hear the First Minister and I would be grateful if colleagues could desist from commenting on that. To final points, I suspect that the Conservatives will also not want to hear. It is the case that on the issue of living standards, the director of the Institute for Fiscal Studies said yesterday about the budget that this is actually awful.  This is what the Resolution Foundation had to say. Of the 4.4 million households in universal credit, around three quarters will be worse off as a result of decisions to take away the £20 a week uplift. Beyond the headlines, those are the realities for individuals and families right across the country. That's why the Conservatives didn't want to hear the facts. I call Liz Smith. Could I ask the First Minister what the Scottish Government's reaction is to yesterday's call from the Scottish Tourism Alliance to extend the period of business rates relief in retail hospitality and leisure on a similar basis to that that was announced by the chancellor yesterday? Of course we already had extended 100 per cent rates relief when the chancellor didn't do it for other parts of the UK. There is 100 per cent rates relief already in place in this financial year. We will bring forward our budget in due course for scrutiny by this Parliament and I can give a guarantee that we will be fair to the retail sector as we have been in a way that the UK Government over recent months has not. Maggie Chapman. Can I ask the First Minister what her reaction is to the aviation cut in yesterday's announcement? What demand management can we put in place to ensure it does not encourage climate polluting frequent flying within the UK? First Minister. It was the wrong choice a few days from COP26 starting when all of us have an obligation to think about how we contribute to reaching net zero in saving the planet that the chancellor chose to make that cut in aviation due to its not a choice this Government would have made but it is for them to defend that in the months to come. Question 5, Pauline McNeill. To ask the First Minister what the Scottish Government anticipates the impact of COP26 will be on Glasgow's healthcare services. The Scottish Government is working very closely with NHS Greater Glasgow and Clyde. We have been doing so for some months to plan and prepare for COP26. Information from previous summits tells us that the impact on routine health services is not substantial, however we are not complacent and we also recognise the risks associated with hosting COP during the pandemic. We have been working therefore with the UN, the UK Government, Glasgow City Council and Public Health Scotland and others to put in place measures to mitigate those risks and the potential impact on the NHS and other public services. Arrangements are in place to closely monitor Covid cases and respond swiftly and appropriately to any increasing cases in order to minimise impact on healthcare services. Pauline McNeill. Like the First Minister, I would like to recognise the hard work that has been done by our health services but I hope that you might share my concern the health board papers this week. Yng nghymru, Glasgow and Clyde said that no specific provision has been made for additional inpatient capacity during COP26. I hope that the First Minister will note that it is not really comparable to the G7 summit because there will be around 14,000 delegates a day at COP and the activist march on 6 November has been licensed for up to 100,000 people. It is obvious that given the Covid cases are high and hospitals are already under or overwhelmed, there is likely to be an increase in patient numbers. In light of that, can the First Minister today or at some other time tell me what will be the receiving hospital for COP26 and will that particular hospital have additional capacity? Finally, the First Minister will have an interest in her constituency in so far as I hope that she could tell me that there will be safe and guaranteed routes to accident and emergency, not just for ambulances, but for the general public who will need to attend A&E during COP26. I hope that she could give me whatever assurance that she can, that she will be mindful of the need to constantly ensure that these safe routes to accident and emergency in her hospitals is provided. Last point, which is of course extremely important. All of that is factored into the transport plans that are in place as people would expect it to be. On the broader question, Parliament will mention G7. I know that Greater Glasgow and Clyde have been looking at previous COP summits to try to assess the likely impact on routine healthcare. I think that it is important to recognise, though, that previous COP summits have not taken place amidst a pandemic, so there may be a different impact here. There is a number of contingency arrangements in place. Both I and the Health Secretary have been looking closely at the contingency arrangements that Glasgow health board have in place. There is substantial on-site health provision to try to reduce the impact on hospitals in Glasgow. Greater Glasgow and Clyde health board is increasing the staffing that will be in place during the summit. Of course, the hospitals that will receive patients will depend on the nature of why patients are being admitted to hospital and, of course, the level of impact across the city. There are substantial contingency arrangements in place, and I hope that that will give some reassurance to Pauline McNeill and the chamber. With significant road closures and ill-time roadworks all across Glasgow, what provisions have been put in place to allow hard-working NHS staff to get to work? First Minister, the transport plans that are in place take account of all of these things. They have been communicated to people over quite a significant period so that people can make arrangements. It is vital that those who work in our health service get to work and that has been part of the consideration as these plans have been put in place. Question 6. Mark Ruskell. To ask the First Minister what the Scottish Government will be doing to amplify the voices of the global south at COP26. First Minister, we have committed to ensuring that our programme for COP26 is inclusive and that we work to amplify the voices of those who too often are not heard, including through the Glasgow climate dialogues to learn, listen and engage on the key issues for those from the global south. We want people to be at the heart of decision making in Scotland and at COP, which is why we also supported the global citizens assembly to bring lived experiences of global citizens directly to COP. We are amplifying youth voices in Scotland and Malawi for example through our Malawi climate leaders programme. I expect to have a number of meetings with representatives of the global south over the next two weeks to listen to their perspective on the climate crisis and to hear how we can further support their voices in their asks of COP26. Mark Ruskell. I thank the First Minister for that answer. It's clear that leadership is being shown by the Scottish Government here, but the global south has been clear that a just transition away from fossil fuels, which leaves nobody behind, must also be a priority at this COP. Let's look at what the first nations of around 5 million people are doing using their full powers over energy. New Zealand, Denmark, Ireland, Costa Rica are all moving on from the era of oil and gas. The case for independence rests on Scotland proudly joining them as a world leader in this real just transition. Does the First Minister agree? First Minister. Yes, we're all in a transition away from oil and gas and as I said earlier this week, we have to accelerate that transition as far as possible. That's why, for example, the Scottish Government has commissioned new research to look in detail at how we can do that and how quickly we can do that. We've got to build up the alternatives, we've got to do that as quickly as possible, but crucially we've got to support those who work in oil and gas transition into the jobs of the future. The Scottish Government is very focused on doing that and to make sure that the just transition is one that is fast enough for the future of the planet. Karen Adam. At the most recent international development CPG meeting we discussed those people who are the most disproportionately affected by climate change in the global south, such as young people, women and the marginalised. It was stressed that any international aid must get to the local communities because they are best suited to ensure that the aid is distributed where it is most needed. First Minister, what steps can the Scottish Government take to ensure that aid is reaching those local communities and vulnerable groups? First Minister. In trying to amplify those voices over COP, this will be one of the key issues. Climate finance is one of the significant strands of the discussions that will take place in Glasgow over the next two weeks. The quantum of climate finance but also how that finance is used so that it's not just on mitigation but on adaptation and crucially on what's called loss and damage. Our own climate justice fund is looking to pivot towards that. We can lead by example in what we do but also do everything we can to make sure that the voices of the global south are heard on those issues and that's exactly what we intend to seek to do. Question 7, Willie Rennie. To ask the First Minister what the Scottish Government's position is regarding the use of facial recognition technology in schools. Facial recognition technologies in schools don't appear to me to be proportionate or necessary. That said, as Willie Rennie will be aware, the introduction of biometric identification systems is a matter for local authorities and schools. The Scottish Government is clear that prior to introducing biometric systems and education authority should carry out a privacy impact assessment or proportionate equivalent and consult with pupils and parents. Information should also be provided on how to opt out consent issues and alternative systems that may be used. Schools and local authorities will also pay due attention to the information commissioner's office requirements that organisations using facial recognition technology should comply with data protection law before, during and after its use. Willie Rennie. The subtle change of tone from the First Minister is mildly welcome but I'm afraid she's not doing enough on this because it's about children's rights and she doesn't seem to be bothered about it. The United Kingdom Information Commissioner has called a halt to the scheme in North Ayrshire but it should never have got this far. The SNP Government previously opposed the Scottish Biometrics Commissioner having a role in health and education. If the First Minister won't intervene on facial recognition will she at least now support the expansion of the remit of the Biometrics Commissioner so that he can intervene? I'm happy to give consideration to that suggestion but of course the rights of children are hugely important to all of us in this chamber. We're committed to ensuring that their rights are protected and that includes the right of children and young people to the fair and lawful process of their personal information. I'm not aware of any subtle change in the tone of my answer what I set out in my answer is important. It recognises the role of local authorities when we're talking about schools but also sets out the requirements that local authorities need to pay heed to. I think that's the appropriate way to deal with what I accept is a difficult and sensitive issue. I call Pam Duncan-Glancy. Thank you. The First Minister, like me, has the privilege of representing Glasgow as an MSP. I'm really disappointed to hear her response today to Anna Sarwar's question. Asking you to meet with cleansing workers tomorrow because of the state of our streets and because there are rats running about our streets is not talking Glasgow down it is letting Glasgow down. As my representative, as my MSP, I ask the First Minister again please meet with cleansing workers tomorrow and hear from them firsthand what is happening. I can assure you that there are rats in our streets there are rats in my flat as my representative, please meet with cleansing workers tomorrow and show that you care about Glasgow. First Minister, I think that people who live in and work in Glasgow certainly in my constituency know that I care deeply about all of the issues that they face that presumably is one of the reasons that they have elected and re-elected me as their representative. I listen to cleansing workers I listen to people across Glasgow on a daily basis. My job as a local representative is to represent those interests which I do every day to the best of my ability. My job as First Minister is to try to find the solutions to these issues that's why the Scottish Government will continue to do everything we can to support COSLA to find a solution to these issues. I don't shy away as a resident of and a representative of the city of Glasgow I don't shy away from the challenges that the city faces but I do think some of the language that Labour is using about Glasgow some of the ways in which Labour is seeking to characterise the city of Glasgow is doing a disservice to the city and to people who live there and they're doing that for political purposes not in the interests of the city. Thank you. That concludes First Minister's questions. Point of order, Finlay Carson. My apologies for my late notification, my point of order. I seek your guidance over comments made by Graeme Dey, the minister responsible for transport relating to future rail services in the south-west of Scotland. Speaking in the chamber on 22 September in response to a question from Emma Harper, the minister stated that there will be six additional services operating on the air to Sronarline. I then raised a supplementary question on his information and Mr Dey responded in his haste to jump up and ask a supplementary question, Finlay Carson was clearly not listening to my first answer so I will repeat what I said. It is proposed that there will be six air to Sronarline services compared to the current timetable. Subsequently, I asked Spice to confirm that the information provided by the minister was indeed correct. Spice has since confirmed that the minister spoke in error. In the response, Spice stated and I quote to confirm that rail officials in the minister are aware of the error below. The other resulted in the analysis of figures by rail officials which were then provided to Mr Dey. Despite that fact and apparently the minister has been informed that I have received no communication or apology from Mr Dey in this matter I am therefore seeking your guidance. Has the minister made any attempt that you are aware of to correct the record on this matter now that you must surely be aware of the error that it made and if not, given how long the misinformation has been part of the official report can you advise how this error should be addressed and whether there has been any breach in the code of conduct ministerial or otherwise? Thank you Mr Finlay for his contribution which is now on the record. The member will be aware that the content of members' contributions is not a matter for me in the chair. There does however exist a mechanism by which members can correct any contribution that they have made.


And ahead of the 2023 general elections, Governor Amadou Fintany has urged Nigerian politicians to play politics according to the rules. He made these known today at the 8th ground in Yola to mark this year's Ido Fittri celebration. All contributions will be in tackling the purity challenges with the care of Allah the society will be free from all forms of evil.  It cannot be done. So therefore we are calling for all people on this point, just not bazaar, that we should make sure that we impress peace. It is therefore advised on the slaves to practice what they have learned through the amount of Ramadan. The amount of Ramadan is the amount of forgiveness. If you seek forgiveness from Adnan, from Hautea, they are doing throughout the league and it is quite sure and commendable for you to also forgive you.


Welcome back. So, we were discussing Borel sets, we bring to discuss Borel sets, Borel sigma algebra. So, we said that, so we were looking at the 0 1 interval, I took omega equal to 0 1. So, why I took open here and close here, we will do not worry about it so much. So, at this point, so I took this uncountable sample space omega equal to 0 1. And I, so one thing I said is that, if you consider all possible subsets of omega, it is too complicated, too large sigma algebra to assign even a uniform probability measure on. Remember that, it is an impossibly result. So, we started out saying that, we cannot do with such a large sigma algebra 2 power omega. So, we need to settle for a simpler smaller sigma algebra. So, we said that, the sets of interest to us are in fact, the open intervals. So, we wanted to keep intervals as being interest, sub intervals of 0 1 being interesting. So, we said C naught is the collection of all open intervals, intervals contained in omega. So, I took this collection and then I said, the sigma algebra generated by this open intervals is your Borel sigma algebra. In other words, the Borel sigma algebra is the smallest sigma algebra containing all open intervals. So, if there is any sigma algebra on omega, which contains the open intervals, the Borel sigma algebra is a sub sigma algebra of that. Is this clear? So, now and elements of, so this is denoted by scripted B, because it is a collection of subsets. We have to, so and elements of this B are called Borel sets. It is a simpler set. So, what is the Borel set? It is an element of the Borel sigma algebra and the Borel sigma algebra is the sigma algebra generated by open intervals on 0 1. Now, this Borel sigma algebra is, it turns out that it is not 2 power omega. It is a much smaller sigma algebra than the collection of all subsets of 0 1, which means there are subsets of 0 1, which are not Borel sets. In fact, what can be shown is, it has been proven that the set of all Borel sets, there are only as many Borel sets as there are real numbers or it has the same cardinality as that of the continuum. Whereas, 2 power omega has a strictly bigger cardinality. So, it is a fairly small sigma algebra, this Borel sigma algebra, but on the other hand what is nice about it is that it contains practically all sets, which are of interest to us. In particular, it contains intervals of course, and it contains also many bizarre sets, many interesting sets are included in the Borel sigma algebra. Although the Borel sigma algebra is a much smaller sigma algebra than 2 power omega, it will be very difficult for you to think of a set, which is not a Borel set. There exist such sets, there are lots of sets of subsets of 0 1, which are not Borel sets, but anything that you can normally, you and I can normally construct is likely to be a Borel set. So, it is non Borel sets are very bizarre, they are only of academic interest by enlarge. So, anything like this is one thing we will prove immediately, we will prove that the singleton set, the set only containing one real number in 0 1 is a Borel set. So, let us prove this proposition, let B be in omega, so B is in 0 1, then the singleton B is a Borel set, Borel measurable set or Borel set. So, how do you prove that some set is a Borel set, see the Borel sigma algebra is generated by open intervals in omega. So, the one thing you know for sure is that all open intervals are definitely Borel and what is. So, what is if you want to prove that some other set is some set, which is not an open interval is a Borel set, what do you want to prove, what do you have to prove? It should be either a it should be expressible in terms of either compliments or countable unions or countable intersections of the generating class, which is the open intervals. So, if I have to show that the singleton is the singleton set is not an open interval, it is just has that 1 point B. So, I am looking at this 0 1 interval and I am taking some point B, this is the singleton set containing that B alone is my set. So, I want to prove that is Borel and the way to do that is somehow express the singleton as a either a compliment or countable union or countable intersection of open intervals, which is the generating class. So, there is a standard way of doing that let me do this correctly. So, you can write the singleton B, you can verify this, you can write the singleton B as intersection N equal to 1 to infinity B minus 1 over N open interval B minus 1 over N B plus 1 over N. So, you buy that. So, I am saying that this set is equal to that set, which means I have to prove B is contained in this intersection and nothing else is contained in this intersection. So, in each of these open intervals B is certainly contained. So, the intersection certainly contains the element B that is very clear. So, if you want to prove that no other element let us say some other element H is not contained in this intersection. So, you will write H. So, you will if B is different from H you will at some point N will become. So, large that B plus 1 by 1 will exclude that H, if even if H and B are very close together at some point N will become large enough to exclude that H. And therefore, you can prove that no other element is contained in this intersection. So, there is one little thing I have. So, this you will buy I think right. So, are you ok with this intersection any questions on this this think about it is not. So, difficult you just have these are standard things you should get used to doing. So, the one little problem here is if I do let us say I take N equal to 1. So, this intersection this interval will be B minus 1 to B plus 1 and this may go outside omega. So, just to be certain I will intersect all these guys with. So, now these are all subsets of omega which are all. So, they are all intersections of. So, I am intersect omega is definitely borel because say the subset has to be in the sigma the sample space has to be in the sigma algebra. And these guys are all intersecting generating class open intervals with omega this. So, this whole thing is borel measurable right. And therefore, countable intersections of these borel measurable sets are in fact borel measure question. So, no no no. So, this intersection. So, I want to make this perfectly clear this is not a limiting operation I said this many many times. So, this is not to be understood as a limit of a finite union. This consists of all those elements which are contained in each one of these sets. You are not you are not limiting N to infinity you are not taking a finite intersection and limiting N it is not what you are doing. You are considering this sequence of sets you can call it B N or whatever you want. I am you are considering those elements which are common to each of these B N's. And the claim is that the only such element is little b all open intervals which are contained in 0 1 no no no. So, I am taking C naught. So, that is only for probability measure that is but you want to have closure under all countable unions or intersections. And I am taking the generating class to be all open intervals contained in 0 1. Now, all see C naught contains all those open intervals which are subsets of 0 1 that is it all of them. So, I am saying that see there are 2 cases right. So, you understood why I intersected with omega because if my N is very small like 1 or something this set may actually be bigger than omega. So, I just want to talk about subsets of omega. So, I am just intersecting with omega whenever my set is whenever this guy is going outside my sample space that is just a hack. But otherwise these guys are subsets of open intervals in 0 1 they are all Borel sets. So, if you intersect it is omega also you will get a Borel set right by definition omega is Borel measurable is not it. So, this so this intersection that is a Borel set. And I have a countable intersection of Borel sets must be a Borel set because it is a sigma algebra right. So, I have expressed my single term as a countable intersection of sets which are certainly Borel sets. And since the Borel sigma algebra is a sigma algebra I have to this must be a element of the sigma algebra. And therefore, the single term is Borel alright is this clear. So, next so the next question is this to. So, if from this you can so for example, you can next prove that. So, proposition you can prove that A B a set like that right where A and B are both contained in. So, this where this interval is contained in 0 1 is a F measurable set. So, this A B closed A B closed A open on that side are Borel sets. Why see I only said that see open interval A B is a Borel set by construction. But I can write. So, if you want to prove that that is a Borel set what would you do? You can simply. So, I have proven that the single term is Borel. So, I can write this as open interval A B union single term B. So, it is a union of two Borel sets is a Borel set. And similarly, this and this or if you want to do it from first principles you can still do it right. You can write A comma B let me do this correctly is equal to intersection n equal to 1 to infinity A comma B plus 1 by n open right. So, you can do that trick also if you want to do it from first principles these are all Borel sets because they are all open intervals. But the intersection what will happen is B will be included in all of them, but nothing beyond B will be included you can prove that again right. Of course, I have to intersect with omega right just to be sure that B plus 1 does not go outside omega right. So, this is a standard trick is this clear or of course, you can just take union of single term with open interval A B that is also fine right. So, you have so all these intervals are now Borel sets right. And anything which is a countable union of these any of these intervals or countable intersections of any of these intervals complements of any of these intervals they are all Borel sets. In fact, the Borel sigma algebra also contains some fairly complicated sets. So, we remember we encountered canter set right. So, the canter set. So, we defined it as the set of all subsets of 0 1 set of all numbers in 0 1 whose ternary expansion can be written without a 1 in it right only 0s and 2s right. So, another way to more geometric way of looking at it is to you start with this guy. So, you start with this 0 1 interval and then you will take the mid open third and remove it. In fact, you can show that all numbers here have start with the first trit the first ternary bit whatever is will be a 1 and these guys will have 0 this guys will have 2 right.  So, you will be left with only 0 1 by 3 and actually you have to I mean canter set is defined on the closed interval 0 1, but it is not such a I mean it is not such a big deal. And then you will do the same thing again right you will knock off the middle third of those right knock those off and you keep that right. And then you keep doing this right again knock off the mid third of this mid third of this mid third of this and so on right. So, you are successively removing open intervals right and the canter set k is the set of points that remains right that is the equal in geometric way of looking at it right you can. So, this canter set contains it will not be empty ok. So, it is not like you remove everything at some point right it has for example you can show that 1 by 4 is a element of canter set it is not an empty set. In fact, we showed that the canter set is there is infinitely many point it has uncountably it is an uncountable infinity of points on 0 1 correct. Because, it is in bijection with 0 1 power infinity right good. So, this is. So, that is the canter set now this canter set is a fairly complicated set right it has a certain interesting property of a fractal self similar nature. So, no matter how much you zoom in to the canter set it will look the same it will have this structure that middle third will be missing right no matter how much you zoom in it will have this middle third missing and these 2 intervals and you zoom in further it will have the same structure right. So, this has a certain fractal structure. So, it is a very interesting and somewhat complicated set on the 0 1 interval right it does not have any simple structure as an interval or anything it is a very complicated scattering of points right. In fact, you can show that the canter set does not contain any interval because you will chop the middle third if it did right. So, this canter set is in fact, a Borel set you will prove that in your homework how will you prove that. So, you have to prove that either this canter set is can be expressed as a countable union or intersection of any of these intervals actually if you look at it is actually fairly simple and the complement of the canter set which is a set of all things that you remove right. That is a countable union of open intervals because the first stage I knock off that open interval, second stage I knock off 2 open intervals and so on right. So, the complement of the canter set if you canter set is k the complement of k will be a countable union of open intervals which you know to be Borel. So, k complement is Borel so k is Borel. So, that is how you argue these things. So, even fairly not just intervals it also contains certain fairly complicated subsets of the 0 1 interval canter set being 1 example right. In fact, in order to construct I said there are subsets of many subsets of 0 1 exist which are not Borel right. So, in order to construct as I said you have to work very hard it is very difficult for you to think of a set which is not Borel set, but they exist actually. So, if you want you can look at there are examples of. So, let me so extra reading so I will just mention a few resources. So, this is not so I am just putting it as extra reading just for your curiosity. So, this is by no means something that I will hold you accountable for. So, this extra reading so I will first put non Borel sets certain is the most famous examples in terms of a continued fraction there is a certain set of continued fractions you can actually look it up on Wikipedia just search Wikipedia for non Borel sets it has a certain example of a certain collection of continued fractions and the set you can show to be non Borel. And there is an example of a so just going a little bit more crazy there is a set called a Vitale set which is non measurable which I mean this is not really immediately relevant to us, but this is also very crazy set on which you can assign no measure to the set is an example of a non measurable set. So, this is a non Borel set this is a non measurable set. And finally, while talking about these crazy sets probably the one of the craziest of all is given by this paradox called Banak Tarski paradox. This Banak Tarski paradox is a very bizarre paradox where you take a sphere or a just a three sphere. And then what you can do is you can divide up the surface of the sphere into some let us say five parts in fact five is enough. And you can reassemble these five parts to form two such spheres of the same size it is possible it seems very bizarre.  And those five parts you can reassemble back. So, as to get two spheres of the same size the reason all this is happening is because of this bizarre sets called non measurable sets. I have in the notes that I uploaded there is a link to this Banak Tarski this is a very bizarre phenomenon. So, this is all stuff that you will not be accountable for, but certainly will find interesting there is some really bizarre subsets of 0 1 or the surface of the sphere. And they are all they are not these nice sets they are all these non measurable sets to which you cannot assign any measure like length or area or probability or any such thing. So, if these sets were in fact measurable if you then you would can do bit of contradiction how can you have the same surface area doubling.  So, that is the most you know an appealing way of proposing this paradox I guess it is all very bizarre because this pieces are very weird scattering of points which are non measurable. So, these things happen. So, that is for your own edification. So, but I hope you understand these Borel sets. So, what did we say Borel sets they include all intervals and countable unions countable intersections and complements of intervals. They also include certain complicated sets like the canter set and it also I mean any pretty much any set you can reasonably think of without working too hard is very likely to be a Borel set that is. So, it contains sets all sets of practical interest to us. So, if you want to construct a non Borel set you have to work very hard to construct one is this clear. So, this is the Borel sigma algebra which is a very small manageable sigma algebra on the 0 1 interval, but which nevertheless contains most sets of practical interest. So, are measurable space from now on will be what omega comma B I will only take. So, omega is this guy and I will take. So, measurable space that I will take from now on is omega B 0 1 when the context is clear I will not write B 0 1 when the context is not clear I will write what Borel sigma algebra. So, for we started off.  So, now aim. So, now my aim. So, what is the initial aim we started out with I want to assign a uniform probability measure on 0 1 I said I could not do it on 2 power omega. So, I settle for a smaller sigma algebra I just kept the intervals and generated a sigma algebra with those intervals and that I ended up with B as my sigma algebra. Now, the question is can I assign a uniform measure on this Borel sigma algebra on only on Borel sets. So, essentially if the answer is yes you can do it now it is a smaller sigma algebra and I can assign a uniform probability measure on omega B. So, I will assign probability measures to only Borel sets. So, there are many subsets of 0 1 like all these crazy servitale and non Borel sets right those guess I will not assign any probability measure 2 I will only worry about Borel sets. So, under the measure that I am uniform measure I am going to put only Borel sets will have a concept of a length or any probability measure on it for that matter right non Borel sets you cannot ask me what probability it is or what length it is it is meaningless to speak of this thing just like it is meaningless to speak of the area of these guys right. So, now I have to start telling you how to put this uniform measure on omega comma B are there any questions at this point. So, I want the following rate I want to put a uniform probability measure on omega comma Borel right. So, my omega is 0 1 I have the Borel sigma algebra on it I want to put a uniform measure on uniform probability measure on this space this measurable space right.  So, I want. So, if I give you a nice Borel set let us say an interval from A B say let us say open A closed B right I know what measure I want for those sets I want it to be B minus A right the length of it right that is really what we started out saying right I want the property that the measure of the interval A B should be something proportional to B minus A and I also said it should be translation invariant right. So, I will start of saying that for the generating class or for all these nice intervals I want the measure to be the length of the proportional to the length of the interval. In this case it you can take it to be equal to the length of the interval because I mean the length itself is 1 and you will automatically get a probability measure right. So, I want to say that for these generating classes my measure should be proportional to the length, but that is not enough right I have to specify the probability measure for all Borel sets which can be fairly crazy right I mean for if you have something crazy like canter set right how do you it is not a it is not like a collection of intervals right it is a very weird scattering of points, but the measure I assigned should hold for all Borel sets, but I can only specify a measure for these nice intervals right. So, I have to be able to extend this measure right this measure I assigned to these intervals to all these crazy Borel sets also right. So, this is what our program is now. So, I will do the following. So, I want to take so I want to start with a very simple collection of subsets. So, let us again take omega equal to 0 1 as usual and I take f naught is the collection of subsets of omega which are finite unions of disjoint intervals of the form a B plus the empty set. So, I am saying so I am taking a collection of subsets of 0 1 which contains the following subsets it contains the null set. So, put the null set in so you are collecting subsets of 0 1 put the null set in first and then you put all subsets which are of the form open a closed B or finite union finite unions there of not countable just finite unions there of.  So, f naught consists of sets which are finite unions of these semi open semi closed intervals it also contains null set. So, now what I am going to do is I am going to say that if I am if you give me an element of f naught right some set like that I am going to say it is probability measure I want it to be equal to b 1 minus a 1 plus b 2 minus a 2 plus so on to b n minus a n right. But I want want to make sure that that measure also extends to all boreal sets right also to countable union this is only a finite union remember right. So, that is our program that is what we have to do. So, now, so the extension of this measure right I know what the measure of these guys should be right b 1 minus a 1 plus b 2 minus a 2 plus so on. But I want to extend that measure to the entire boreal sigma algebra all boreal sets. So, the way you do that is by a very fundamental theorem in measure theory known as Karate Odery's theorem Karate Odery's extension theorem Karate Odery's a British mathematician I will state the theorem without proving it. So, that is the main technical tool that helps us extend are what measure we want on sets of this form to all boreal sets. So, before that I want to get these certain properties of f naught I want to finish. So, I want to say that f naught is first of all an algebra. In fact, the moment I wrote down some f naught you probably guess it is an algebra and I mean because that is the notation I have been using all along right. I use f naught for algebra and f for a sigma algebra right I did not tell you that at this point right I will just say f naught consist of a certain kind of sets. But you can prove that f naught is an algebra how do you prove that f naught is an algebra it contains phi by definition it has null set then you have to prove complements are also sense a f naught. So, if you have some set like. So, open closed open closed right this is 0 1 and I have some set like open closed open closed then the complements will also be of that form right this guy the complements is also open closed open closed and so on right is omega in f naught the sample space is it an element of f naught it is right because the which is now you see why I took open closed here that is also an element of f naught right. So, all these simple properties hold and how about countable unions sorry finite unions f naught should be closed in the finite unions which you can show right it is a. So, it is a fairly simple homework problem I leave it you as a homework problem. So, this I leave you as a homework I just spoke out the proof you just have to write it out. So, closure under complementation and finite unions now this f naught is not a sigma algebra. So, this collection f naught that I have defined is an algebra, but not a sigma algebra. So, here is another example of a collection of subsets of 0 1 which is an algebra, but not a sigma algebra you already encountered such an example in your homework I think of some collection which is an algebra, but not a sigma algebra right. So, this is another example is f naught this collection is not a sigma algebra although it is an algebra how do you prove that. So, in order to prove something is not a sigma algebra you have to see the complementation will I mean that will not going to help you because I mean it is an algebra already right. So, you the only way it cannot be a sigma algebra is countable unions are not closed right. So, you can give a very simple example if you take let us say you take sets of the form let us say you take a n sets of the form 0 comma n over n plus 1 that. So, that will do I think you do that n equals 1 2 dot dot. So, a 1 will be 0 half right a 2 will be 0 2 over 3 closed and. So, this is the sequence of sets I have are these guys in f naught. So, a n's are in f naught for all n agreed why because they are of that form know they are of this form. Now, if you take countable union countable union n equals 1 through infinity 0 comma n over n plus 1 what you get 0. So, I am taking countable union not countable intersection then I am taking countable union of 0. So, I will the first set will be 0 comma half second set will be 0 comma 2 by 3 and so on they are getting bigger and bigger right. So, I mean so the point is this right. So, this union contains now you have to look at. So, you are looking at the set of all points contained in at least one of the a n's right correct. So, you can show in fact this is equal to open interval 0 1 this is open you can show that these two sets are equal. So, the open interval 0 1 includes all points in 0 1, but does not include 1. So, you can show that this none of these a n's include 1, but for any number 1 minus epsilon 1 minus a little bit will be in one of these a n's. So, this you think about this is fairly easy I mean just try and prove it vigorously. So, you get this union is equal to open interval 0 1 is open interval 0 1 in f naught no it does not it is not in f naught, because you only have open closed on this. So, this is a not an element of f naught correct. So, here is a concrete example that my f naught is not an sigma algebra although it is an algebra. So, proof of two finally, I want to state the third property that I am going to look at the sigma algebra generated by this collection f naught. So, f naught is sets of these forms collection of sets of these forms right. So, I am going to generate the sigma algebra. So, I am going to consider the sigma algebra generated by the f naught. So, I told you f naught is not a sigma algebra right. So, I have to make it into a sigma algebra right make it into by which I mean consider the smallest sigma algebra that contains all elements of f naught which is what this is sigma f naught. Any guesses on what this will be no see f naught is empty set and sets of this form right. I told you with an algebra it is not a sigma algebra, but in probability theory I want to work with sigma algebra. So, I am considering the smallest sigma algebra generated by f naught what it turns out is that this sigma algebra is no different from the Borel sigma algebra it turns out this is equal to b 0 1 this you can prove. So, what am I saying in effect what am I saying see the Borel sigma algebra initially was defined as the sigma algebra generated by open intervals. I am saying that even if you generate a sigma algebra by the semi open semi closed type of intervals it is still be the same Borel sigma algebra. Meaning that the sigma of f naught is same as sigma of c naught which is the open intervals. So, I have not proven it yet right. So, do I have the time to prove it I have 5 minutes let me try and prove this. So, how do you prove that. So, how do you prove something like this how do you prove that the sigma algebra generated by these class of sets is no different from the sigma algebra generated by open intervals. See first of all you should recognize that this is an equality of 2 what kind of objects collections right. So, this is a collection of some subsets this is some other collection of subsets of 0 1 and I want to prove that this is equal to that which means I should prove. So, this is contained in that and that is contained in this right that is what I need to prove. Now, which is the easier part. So, this let me just write this this guy is nothing but sigma of c naught by definition c naught is open intervals. So, this is you already know that. So, how do I prove that. So, I want to prove this is contained in Borel and each Borel set is contained in sigma f naught right. So, let me try and do that proof of 3 to show sigma f naught is contained in b 0 1 it is enough to show f naught is contained in b right why is this. So, this is the smallest sigma algebra containing f naught. So, I am saying that if the Borel sigma algebra is a sigma algebra containing f naught then it must follow that sigma f naught is contained in b by the theorem right with the theorem is generated about sigma c j j generate a sigma algebra. So, you buy this. So, if I prove this I prove this why because here is a sigma b is a sigma algebra containing f naught.  So, in order to prove this I should prove that every set in f naught is a Borel set right and what is the typical element of f naught it can either be an empty set which is definitely Borel or it must be a set of this form right, but I know that half open half closed intervals are Borel. So, finite unions are Borel right. So, this is clear. So, this is easy right this you this is fairly easy I just argued it out prove that element. So, empty set is in b and also any element of f naught of this form you prove an already that they are Borel sets right. So, this follows if this is true then this is true. So, one direction is over the one direction is very easy right. Now, I have to prove which way I have to prove that next to show which direction B is contained in sigma f naught to next to show we proceed as follows. So, if you take any open interval a b which is a subset of omega open interval a b can be written as union let us see union n equals 1 through infinity a minus 1 by n is that correct just like this know I mean something I am writing something similar to this right. So, you agree with this equation verify this. So, now what have I shown any open interval is a countable union of elements of f naught right. So, these guys are all elements of f naught correct. So, countable union of elements of f naught will be a element of sigma f naught certainly right. So, what is this prove this implies every element of c naught is an element of sigma f naught correct. So, I am doing this proof I mean just as a practice for you I mean I think you should these you will encounter several such proofs. So, your so this implies c naught my collection of open intervals we certainly contained in sigma f naught. So, why is that true for any open interval a b I have written it as a countable union of elements of f naught. So, this countable union must be an element of sigma f naught fine everybody with me. So, then this follows correct now what this one step remains. So, I have identified a sigma algebra that contains c naught correct. So, therefore, sigma of c naught must be contained in sigma of f naught what is sigma of c naught borel right this must be borel. So, I proved it both ways right. So, I proved that the sigma algebra generated by sets of the form f naught is nothing, but the borel sigma algebra. In fact, in the homework you will prove that if you were to generate sigma algebra using closed intervals you will still get borel sigma algebra. So, you can generate borel sigma using open intervals or closed intervals it does not matter this everybody with me. So, this lemma is important. So, lemma has 3 component that says that f naught is an algebra not a sigma algebra and sigma of f naught is a in fact, the borel sigma algebra. So, we will use this lemma very heavily to define the uniform probability measure or the Lebesgue measure on the 0 1 interval which will be topic of tomorrow's lecture. Thanks.


Hello everybody, welcome back another AI video in this one. I'm going to show you how to learn anything you want with chat GPT and better yet, I'm going to show you how to do it visually. That's right. You can create graphs, diagrams, things like that, so it can help you sort of better understand it. Now that's a big talk, but if you look here at my previous chat, you'll notice here that I've got, you know, I've used the Pareto principle, which I'll show you how to use quickly. And then it also created an editable graph for me so we can go ahead and go in there and make changes to it. So this is just one example of many. Let's get going. Let me show you how to do this. First up, I'm just going to create a new chat and then you're going to see that I'm in GPT-4. This is the latest version at the time of recording. And then when you click on that little GPT-4 button, you get three options, default, browse with Bing, which is nice, and plugins. I'm going to make sure I select plugins. And then you'll see here that when I hover over it, let's see here, I've got the check mark beside the plugins. Now when I click on it, let's go ahead and click on it. You'll notice here that as I go underneath it here, you'll see this little box. It's kind of hard to see sometimes. And you're going to see that I've got show me selected. That's right. Show me the money. Show me the money. Show me the money. Here we go. So I've got the blue check mark there, so we're good to go. I can also go to the plugin store. And then if you don't have it installed right now, it's on the popular page on the second one. You just have to click on the install button and off we go. All right. That out of the way. Let's go. What I want to do is I want to learn how to, how does a car engine work? This is just an example. So again, I've got show me selected. How does, okay, show me, I'm going to just use the exact wording. Show me how a car engine works. Use the Pareto principle. Click enter. Something simple like that. Watch this. I'm going to do a little bit of thinking and here it goes. I can create a simplified diagram of how a car engine works. And it's going to show me based on the Pareto principle, even though the Pareto principle doesn't make a whole lot of sense in this situation, but I just wanted to show you just for the giggles. So here we go. It's going ahead and is running it out. Here we go. Let's render this diagram. This is the good part. So it's yow using show me. You'll see here that it selected it here. It's running. If we click on this, it'll show you exactly what it's requesting for you super techie types. You'll see here that you could even see that it's creating the diagram right here. Here's the request. Here's the response. Look at that. And there it is. This is our diagram showing us the different parts of the engines and how they're connected. And better yet, you can even edit this diagram. So while it goes ahead and fills out the remaining parts, how this works, click on this button here or click on this link part of me. Watch this. There it is. That's we can go ahead. We can download this. We have a code on the left side here. So if we wanted to make changes to it, you'll see here the last one says power to wheels. If I don't like that, power to Curtis Pike, which is me, whatever. You see here, I just made a change there and it's already updated inside the diagram. So this is pretty wild and you can copy the code. If you want to add in stuff, you can go ahead and go, hey, you know what? It's got a fuel system, spark plugs, valves, and I want more. So I'm going to go Zed and then I would go cake, something like that. And you see what happens here? I'm adding to the diagram. So this is something that is amazing. Now let's go back here to the back to the chat GPT. You'll see here that it says in this diagram and it just explains exactly how it works. And if I want to learn more, I could be like, explain how the fuel system works. So I'm just picking one at random, hit enter, and then bang, it's going to go ahead and it's going to go through that all again. And it's actually probably going to create me another diagram. So there you go, guys. And if I don't get the diagram, show me how the fuel system works. And I'm going to hit enter. If the diagram doesn't show up, sometimes it does, sometimes it doesn't. This time it looks like it did it. It did it for us already. Look at that. This is amazing, guys. So this is the show me plugin. It's available in chat GPT-4 in the plugin App Store. It's free. Thanks for watching.