 set before, so the coffee break is back at four on the schedule, so the exercise session is from two to three thirty, but since the room is booked until four you can stay over and the coffee break starts at four anyway, and the second one is that the group discussion for women and gender minority is on the terrace, like where we did the ice breakers yesterday and not in the Adriatico Terrace like it was in the program before, okay, so let's start. So I will be talking about measure preserving transformations and before transformations, first I would do a bit of a recap of the things from the lecture from yesterday that I will need in my class. So X for me would always be a metric space or well it could also be a topological space for most of the things I'll say and most of the time it will be either zero one or S1 or the things we've been looking at so far, so from yesterday we know that XB is a measurable space, so X is our metric space and B is a sigma algebra of sets that I remind you that it means that it contains the empty set and if it contains a set then it contains its complement and also if it contains a countable number of sets then it contains its union and those are called the measurable sets. Nice sigma algebra that we often use is B the Borel sigma algebra that is the smallest sigma algebra containing open sets. On a measurable space we can put a measure that is a function from the set of measurable sets into our plus union infinity such that the measure of the empty set is zero and if we have a countable union of measurable sets that are pairwise disjoint the measure of their union is the sum of the measure of those sets and we will call XB with the measure mu a measured space a measure space so a measurable space is a space on which we can put a measure because we have a collection of sets that are ready to have a measure put on and then we had the extension theorem tells us that giving a measure on an algebra set and it's uniquely the measure and the sigma algebra generated by this algebra so we don't want to to say what the measure of every set is but it is enough to give it on an algebra so on finite unions and then it extends to a measure on all measurable sets I want to give you an example of a measure that I will use so if we have a function from an interval to our plus so a positive function that is integrable normal Riemann integrable then if I take a subinterval I can define the measure of the sub integral to be a subinterval sorry to be the integral of f of x the measure of the empty set is is zero because we are integrating over nothing and then the measure of the total space our interval I is finite because f is integrable and if we have disjoint disjoint subintervals it satisfies additivity and so on and so since the the set of finite union of intervals is a is an algebra we can apply the extension theorem which tells you that mua extend to a measure on B that is the Porel sigma algebra okay because the Porel sigma algebra is given by countable unions of intervals and this we call a measure defined by the density f like that because basically f is giving a weight to each point in my interval so I'm saying that it's a density because if we have a space with particles on it for example and the particles are more dense in a part of the interval then this integral will be bigger so given a positive function this can give us a measure on our space okay so this is what I will need for measure theory and now we pass through transformations we fix space x with its measurable sets and its measure so measure space and then we say definition but t a transformation from x to x is measurable subspec subset measurable set then the pre-image of a is still measurable and so this is first definition and second definition t is measure preserving well first we want t to be measurable and second so we want the the measure of t to the minus one of a to be exactly the measure of a for each a that is measurable so what we're saying is that we have our space and we have the measurable set and if we take one of those measurable sets we can pull it back we can take its pre-image in x and we want this new set first of all to be measurable and then once it's measurable we can calculate its measure and it preserved the measure in the sense that when we go back we still have same measure so first some remarks the second point on my definition doesn't make sense without the first because I can't take the measure of the pre-image unless the pre-image is also measurable second this though is not very restricted restrictive because point one is often satisfied in some sense so one is satisfied a large class and to convince you that this is true I will tell you that for example if we take the Borel sigma algebra and t continuous this implies that t is measurable so in some sense there are that are uglier transformations but a lot of natural transformations and in particular the ones we will be we are thinking about mostly this week they all satisfy point one now whether they satisfy point two or not is what we will see in the examples later so a third remark you want to know if we could use the image instead of the pre-image it's quite natural to act so would it be would it be the same thing would it give the same measure preserving transformation so is it equivalent to use measure of t of a equal measure of a for each a and b well the answer is no and we will see it in an example later in my class and but there is a case where this is equivalent and this when t is invertible this is equivalent number two and this will be one of my exercises for this afternoon and then the last remark be remark number four is that what happened if we take a composition so we have a transformation from x to x and then a second transformation from x to x both measurable measure preserving what happened when I compose them well then as compose t is also measure preserving and this is very easy to see just writing down the definition so measure preserving transformation are very nice objects and we will see some examples very soon but first I want to give you a another definition that Oliver briefly mentioned yesterday that is that if we have a transformation we can basically set ourselves up to have a measure preserving transformation so so the map t star mu from B to our plus union unity is the push forward measure and is defined as so we need to say what t star mu is on each measurable set well we say that this is the measure of t minus one of a reach a measurable so we have a measure on on our space and then we want to give another measure on our target space and we pull it back using the transformation and well the first thing to do is to prove that so remarks is that we need this term you to be a measure and this also will be one of the exercises and then well I said that we set ourselves up for being measured preserving and in fact the point is that the issue measure preserving even only if t star mu equal mu and to see this one just write down the definition so the measure of a so to be measured preserving this needs to be the measure of the pre-image and then but then by definition of t star this is exactly t star mu of a and since this is true for each measurable set then this means that mu equal t star mu but again checking this measure preserving property on each measurable set is we always want to try to restrict the things we need to check as much as we can so we have also in this case some equivalent of the extension theorem that we had later that says something similar that says that it is enough to check that we are measure preserving on an algebra more precisely if we have a measure space the sigma algebra is generated by an algebra then the preserves mu if and only if well the property that we want is true for all a in the algebra now instead of in the whole sigma algebra so the proof well one side is easy because if it's new preserving that this is true for each measurable set so in particular is true for the for the algebra to prove the other side we use the extension theorem consider t star mu and mu t star mu restricted to the algebra coincide with mu restricted to the algebra and then by the extension theorem t star mu restricted to a and mu restricted to a extend well here we extend to t star mu here we extend to mu but by uniqueness those two need to be equal and t star mu equal mu is equivalent to say that t is measured per serving and so we're done so now we go to the examples the first example is the doubling bump star for the course from Hannah so we take xp and lambda where x is 0 1 and then B is the Borel sigma algebra and lambda is the Lebesgue measure now we take doubling map t from x to x t of x is 2x mod 1 2x if we are between 0 and a half or 2x minus 1 if we are between a half and 1 so we have the our two branches and now I tell you and I will show you that t preserves the Lebesgue measure so which elements do we need to check new extension theorem we just need to check an algebra and an algebra that generates the Borel sigma algebra is finite unions of intervals so let's first look at intervals so equal R or an interval then and B is the Borel sigma algebra then we want to check it on intervals in R or and so infinite union of interval that is the algebra and then countable union of intervals that is the Borel sigma algebra so let's take our interval AB and cut and see what t minus 1 of AB is well each Y in AB has two per image and one is one per branch so one is on y equal to x so is the point y over 2 and the other is y equal to x minus 1 so x equal y plus 1 over 2 and so this means that the pre-image of an interval is to this joint interval that thing is faster if I re-draw the graph then if I go down there if we take an interval AB here then the pre-images will be this piece here and this piece here so t minus 1 over AB will be the union of A over 2 B over 2 and union A plus 1 over 2 B plus 1 over 2 and so when we go to calculate the Lebesgue measure of this well the Lebesgue measure of this since the union is this joint then is the sum of the Lebesgue measure of the two single intervals and the Lebesgue measure of an interval is so the Lebesgue measure of AB is B minus A and so here we will have B over 2 minus A over 2 coming from the first one plus B plus 1 over 2 minus A plus 1 over 2 so now this half and this half cancel because of the minus so we remain with B over 2 plus B over 2 that is B minus A over 2 minus A over 2 that is A and this is exactly the same that we have here and so it's measure preserving so if it's true on on an interval is true on on a finite union of interval and so by the extension theorem is true for countable unions of interval gave this lemma that says that if it's true on an algebra then it's mu preserving and finite unions of interval is an algebra that generates the Borel sigma algebra I just did it so to prove that these imply this I just say that if it's true for each elements of the sigma algebra then in particular is true for the elements of the algebra that generates it okay no so is contained in B because is the algebra that generates it so measure preserving means that mu of T minus 1 of A equal mu of A for each a in B so in particular is also true for each a in a because a is contained in B and so then that this is exactly this okay so one side is proved like this and the other side we consider the pullback restricted to a and the measure restricted to a and then by the carotidory extension theorem that we that Oliver explained us yesterday we can extend both of those to a measure and so when we extended to these and we extend it to these but by uniqueness they need to be the same and we just said that these imply that it is equivalent to say that we're measure preserving okay okay so here I want to remark this is an example of why we can't use T instead of we can always use T instead of T minus 1 when we define measure preserving so what happens if we use T instead of T minus 1 in the definition of measure preserving well if we consider the measure of the image of a B well if we take already so let's take a B contained in 0 half already here the so so T of a B is to a to B and the back measure of these is to be minus a that is different from B minus a that is the back measure of a B so in this case if we had taken the images instead of the pre images we would have that the doubling map is not measured preserving so the two things are not equivalent and in fact T is not invertible and we said that when it's invertible we can take the images but here is not the case and in fact we would attain a different answer to this question so a second example rotations again we have our x B lambda with x S1 that is the same as 0 1 when we identify 0 1 and then B is again the Borel sigma algebra and lambda is again our Lebesgue then we consider our alpha from S1 to S1 that is the rotation by 2 pi alpha now again we want to check if this is measure preserving or not so where do we check it instead of checking on everything so if x equal S1 and B is the Borel sigma algebra it is enough if we check it on arcs for the same reasons as before so in fact that Lebesgue measure on S1 is the same as the Lebesgue measure on 0 1 with our identification and so we want to check it on arcs and what is the measure of an arc well is the length of the arc divided by 2 pi and we divided by 2 pi so that the measure of the whole S1 is the same as the measure of 0 1 that is 1 because if our circle circle has radius 1 then the length of the whole circumference will be 2 pi so we divide by 2 pi and then we're doing the same thing that we were doing in 0 1 and and now we remark that if that a rotation is invertible and that the inverse is another rotation of angle minus alpha so if the rotation by alpha is counterclockwise the rotation by minus alpha is a clockwise rotation but it's still a rotation and rotations are isometries in the sense that they preserve arc length and so back measure of our alpha to the minus 1 of an arc is the same as the Lebesgue measure of our minus alpha of an arc which is the same as the Lebesgue measure of the arc because this is a rotation okay and so this is also measure preserving and in fact our alpha our alpha is invertible we could also consider the image of an arc and that will still be the same as the measure of the arc itself so in this case it's equivalent to take images but that's just because the rotation is invertible third example so so far we always had Lebesgue measure but we could have also other measure and this is the reason why I gave you earlier the measure with respect to a density so now our third example we take still our interval 0 1 which we very much like and still be the Borel sigma algebra but this time we take the measure defined by the density 1 over x so this means that our function f of x is 1 over x and as I said at the beginning of my class this defines a measure that the measure of a is the integral of array so in this case of 1 over x dx and again since we are in the interval we said that we can just check it on on sub intervals so so first and and now we need to give a transformation so the transformation is the farry map that is well it's called the f from 0 1 to 0 1 defined as follow so f of x is x over 1 minus x if we are between 0 and a half and it's x minus 1 sorry 1 minus x over x if we are between a half and so if we we have our 1 by 1 and we have a half where we have the two branches the map is something like this and so again let's do like before we take an interval up here AB let's take it lower so that we can see better what happens and we see what are the pre-images so so what is f minus 1 of AB so I point y in AB has two pre-images again corresponding to inverting each branch in the first branch we inverted with x equal y over 1 plus y and then the second branch is 1 minus x over over x sorry and then x equal 1 over 1 plus y but now we can see from the picture that if this is this point is out of a over 1 plus a then this is b over 1 plus b but now this one is 1 over 1 plus a and this one is 1 over 1 plus b so t minus 1 of the interval AB is the disjoint union of a over 1 plus a b over 1 plus b union 1 over 1 plus b 1 over 1 plus a so when we try to take the measure sorry this is half of this well we need to use our density so we can we separate the two integrals so is the integral from a over 1 plus a to b over 1 plus b of 1 over x dx plus the same thing between 1 over 1 plus b and 1 over 1 plus a and now we know how to integrate 1 over x so we know what this is so this is so I will so the the measure of AB is the integral between a and b of 1 over x dx that is the logarithm of b minus the logarithm of a because logarithm of x is a primitive so we calculated in the extremities so this now becomes the logarithm of b over 1 plus b minus the logarithm of a over 1 plus a plus the second part that is the logarithm of 1 over 1 plus a minus logarithm of 1 over sorry this is 1 over 1 plus b and now we remember that the logarithm of a quotient is the difference of the logarithms so this becomes logarithm of b minus logarithm of 1 plus b minus log of a plus log of 1 plus a plus log 1 minus log of 1 plus a minus log 1 plus log 1 plus b so we can cancel some terms and this with this and this is log b minus log a which was exactly the measure of AB and this implies so so we checked it on intervals so the same thing can be done on finite unions of intervals and the finite unions of intervals generate the whole Porel's sigma algebra and so by dilemma that we raised f preserves and in fact f does not preserve Lebesgue but if we just write it down then one can see that this is not true but writing the Lebesgue measure here and I think it's time to stop