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y of certain functions as above one might consider whether it s possible to tweak the definitions to avoid this here is a list of reasonable alternatives to the standard order of construction described above in each case i ll draw attention to something that might go wrong focusing on the case but the issues mostly generalise truncation of range we ve touched on this earlier but let s briefly revisit what happens if we define integrals as a limit of intervals over finite range leaving aside the question of whether this is possible for more general spaces see 3 we already have issues with this approach over note that once we restrict to finite range and continuous functions or finitely many discontinuities it doesn t matter if we are using riemann or lebesgue framework but when studying a function like we have truncated results like and so taking the limit would define the integral of f over to be zero while taking the limit would give a different answer so this certainly wouldn t work as a general definition in practice for improper integrals one generally studies with the same restriction on taking as we discussed earlier in this post however this split into the two half lines is very much a feature of the reals even just in there is no easy analogue to this split approximating all functions from below what about if we try to approximate all measurable functions from below by simple functions after relaxing the condition that the coefficients of a simple function have to be non negative well if a function takes arbitrarily negative values eg all values in then it just can t be bounded below by a function which only takes finitely many values and if we allow simple functions to have negative coefficients and support on infinitely many sets this is even worse than the situation discussed in footnote 1 already at this point we would already have simple functions for which the integral isn t defined such as taking for and for it s not going to work very well to study the supremum of a set some of whose values are not defined jointly approximating from below above this is the suggestion that prompted me to write this post suppose we relax the non negativity but not the finite sum condition for a simple function then approximate by simple functions so that and a e that is bounds f from below when f is positive and from above when f is negative the issue here is that it s not clear how to turn this into a definition we can t write because the sup would be attained by considering simple functions g which are zero whenever f is negative and it s not well defined to replace the supremum with a limit it s not a sequence after all if you try and split it as a supremum over the range where and an infimum over the range where then you are actually back to the original definition of the general lebesgue integral and going for a very concrete sequence doesn t help either unfortunately if we define by rounding down up depending on whether is positive negative to the nearest multiple of we could define notwithstanding some of the issues we mentioned earlier about committing to an specific approximating sequence however since the are not monotone in n there is no guarantee that this limit exists and indeed we can construct examples of where the limit does not exist consider for example the intervals and the function so then the approximation and thus which is 1 when n is odd and 0 when n is even with the true definition of the lebesgue integral is undefined which might seem equally disappointing but in reality is more clear than ending up in a situation where you get different limits depending on whether you round down to the nearest multiple of versus rounding down to the nearest multiple of footnotes 1 it s not impossible to exclude such pathologies for example by demanding that the are disjoint but adding constraints to the definition of simple function is far from ideal if we need to work with them for proofs for example under it becomes less obvious that the sum of two simple functions is simple 2 if we define to be rounded down to the nearest multiple of and similarly for then we immediately run into problems when trying to prove that as the rounding down operation doesn t commute with addition 3 and there are measure spaces which are not sigma finite for which e cannot be expressed as a countable union of finite measure sets but we still want a theory of integration over such spaces posted in analysis measure theory university teaching tagged borel measure fundamentals of probability integrable function lebesgue integral measurable function measure theory monotone convergence theorem monotone limit sigma finite simple function leave a reply borel cantelli lemmas posted on october 6 2022 by dominicyeo reply i m currently lecturing my first course at king s which builds measure theory from the ground up to the construction of the lebesgue integral along with some more probabilistic topics in this second week we have been discussing various matters related to the construction of sigma algebras it took me slightly by surprise that this ends up being the natural moment during an undergraduate s progression through probability to introduce the borel cantelli lemmas it s not hard to imagine a pedagogical schedule in which they appear earlier but equally one benefits from seeing them with a bit more mathematical experience even if they are only very loosely related to the construction of the lebesgue integral anyway i noticed that i have not written about these results on the blog before there are a couple of neat applications that lie slightly beyond the remit and time allowance of the lecture course so they are presented at the end context events holding eventually and infinitely often recall that in the notation of probability spaces a sigma algebra provides a set of events for which it is meaningful and consistent to define a probability measure this is extremely uncontroversial in the setting where the set of outcomes is finite but thornier in the setting where is uncountable for example 0 1 or the key axiom for sigma algebras involves countable unions if is a sequence of events in we must also have since the axioms also demand that is closed under taking complements it only takes a few steps to show that countable intersections are also valid ie also it is uncontroversial to note that the main initial motivation for considering the events are that they lie between the intersection and the union in terms of containment infinitely often is relatively uncontroversial just meaning for infinitely many values of n eventually is less clear on an initial reading perhaps eventually always would be better and closer to the title of this blog the meaning is always except for finitely many n which implies that there is some threshold n such that it holds for all n n in set notation infinitely often or i o means that must hold for every n and so the event similarly eventually means that there must be an n for which holds so the event we end up with the richer containment sequence these individual containment relations may not be obvious a priori in this form probably the most clear is the central containment that if an holds for all except finitely many n then it holds for infinitely many n the right most says that if it holds infinitely often then it holds at least once for the left most containment note that is literally the same as the n 1 case of so taking a union over other values of n only increases the set it s also worth thinking about the complements note that and versions of these apply to eventually i o also in particular and so in any calculation or application there is flexibility to study is this is more convenient as we will see it makes sense to focus on whichever of has vanishing probabilities as comment it is standard to write and i am happy to admit i get these confused with non vanishing probability and prefer to stick to eventually i o or the full union intersection statements for safety borel cantelli lemmas the borel cantelli lemmas establish some conditions under which the event almost surely holds or almost surely doesn t hold we ll state them now bc1 if then bc2 if and the events are independent then proof of bc1 since for every n we have since comment we could also introduce the random variable which counts how many events occur ie then and the given condition is equivalent to which certainly implies indeed it s reasonable to think of bc1 as a first moment result with bc2 as sort of a second moment result and this is reflected in the comparative complexity of the two proofs proof of bc2 as discussed it is equivalent to show that to do this we study the intersections of the complements whose probabilities are tractable using independence we have assumed that for each n and so we obtain but since is an increasing sequence of events so taking a union over n is particularly natural precisely by continuity of measure we have comment the independence condition is certainly necessary as otherwise one can have situations where for example with consider for which whose sum diverges while clearly classic examples the famous infinite monkey theorem relatively well known outside the realm of theoretical probability asserts that a monkey typing randomly at a keyboard for long enough will eventually type the complete works of william shakespeare other standard examples involve independent with probabilities decaying at various speeds the latter do have some overlap with the slightly less standard examples presented below example generating random permutations consider a sequence of random permutations where is a permutation of we construct these iteratively as follows is the identity ie the only permutation of one element then for each define from by inserting element at a uniformly chosen position in we ask the question whether this gives a well defined limiting permutation on a necessary condition for this is that the first element is eventually constant but in fact the events are independent and each occurs with probability and so by bc2 on reflection each has the uniform distribution on permutations of so our result is consistent with the idea that one cannot construct a uniform random permutation on an infinite set now alternatively suppose the new element is added at place where are iid random variables the truncation is just to prevent pathologies like trying to add the new entry in place 6 then and since bc1 shows that the first element changes only finitely often a similar argument applies for the second element and the third element and so on consequently almost surely we have a well defined pointwise limit permutation this is known as the mallows permutation on the integers and is one of the more popular models for a random permutation on an infinite set with lots of interesting properties i supervised an undergraduate research project on this model over summer 2022 and will write more when time allows example well separated random integers the final example is a toy version of a construction i ve been working on recently regarding subsets of the vertices of a graph which trap random walk on that graph suppose we are trying to colour some positive integers randomly so that the coloured integers are infinite but also asymptotically sparse we also do not want pairs of coloured integers to be close together let s say that two integers are close if now suppose we colour each integer green independently with probability then whenever we have two integers which are close and coloured green we re colour them both red we would like to prove that infinitely many integers are coloured green let using bc2 we know that almost surely infinitely many are coloured with some colour and it seems implausible that many would be close however the events are not independent so one cannot apply bc2 directly to these however we can study instead we can bound these probabilities as and apply bc1 to show then since we obtain as required posted in measure theory probability theory tagged borel cantelli borel cantelli lemmas emile borel eventually graph colouring infinite monkey theorem infinitely often lebesgue measure mallows permutation measure theory random permutation uniform permutation leave a reply 1a probability extension problems aside this year i was lecturing the first year probability course in cambridge to supplement the usual excellent problem sets i inherited from james norris and many previous lecturers i prepared extension problems for enthusiastic students a handful of the extension problems are original a few are simplifications or examples of problems in graduate level discrete probability the first sheet is more gentle than the subsequent three these can be found below cambridge_extension_problems download cambridge supervisors are welcome to use these in subsequent years indeed anyone is welcome to use these in their teaching solutions are available for instructors please contact me by email to receive these posted on may 3 2022 by dominicyeo tagged 1a probability almost sure convergence birthday problem branching process cambridge covariance discrete harmonic extension examples sheet exploration process extension problems harder problems multitype branching process part 1a polya s urn random permutation random walk size biased snakes and ladders total population size leave a reply bmo2 2022 posted on january 28 2022 by dominicyeo reply the second and final round of this year s british mathematical olympiad took place on thursday here are some thoughts on the problems i wasn t involved in choosing the problems although i did write q4 i ll say a bit more about this geometry problem than about the other three so i ve put this first the copyright to the problems is retained by bmo and i m reproducing here with permission the paper can be found in its original format here problem four let me explain how i devised this problem let a be the point diametrically opposite a on the circumcircle of triangle abc sometimes called the antipode of a now consider any line l through a and let p be the point where it meets the circumcircle again i was particularly thinking about lines l where p lay on the same side of bc as a as depicted because aa is a diameter there is a right angle so i considering intersecting l with the lines given in the question since this would create two cyclic quadrilaterals denoted in the figure let s just focus on this cyclic quadrilateral gives us two equal angles in many different ways but a particularly nice pair is the one shown because acpb is also cyclic which confirms that the angle measure is in particular by extending to e on ac we get a third cyclic quadrilateral and hence find out that so four lines go through d ap bc and the perpendiculars from to the opposite sides of the triangle how to turn this into a question the steps outlined above are relatively clear because the right angle at p is ess...