Ilya
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 Jun 5 comment Euler-like identity for partition function Would it be of any help that $f(x)f(-x) = f(x^2)$? May 4 awarded Popular Question Dec 5 comment Borel class of a set of measures I guess you can show that total variation is a measurable function, but not sure whether it will help to answer which Borel class does $B$ belong to. Dec 5 comment Measurability for disintegration of a kernel Not sure whether this answers your question, but Lemma 2 of "On the existence of good stationary strategies" by Sudderth may be of use. Dec 5 comment Defining density of a random function using Radon-Nikodym Theorem Density is just one representation of a probability measure (through another measure) and it is useful as a tool, when it comes up naturally, not as a goal: by itself it does not tell anything new about probabilities. Now, in the space you've described a natural measure would be the Wiener measure as @Jochen mentioned, but if I'm not mistaken your $P_X$ is singular w.r.t. the Wiener measure. The question is what were you going to do after you'd get a density - if you specify that, I guess the conclusion would be that you don't really need density. Dec 5 comment Property of relative entropy It is equivalent to $P = Q$, see p.7 here. The statement "$P=Q$ almost everywhere" does not make sense. Dec 5 comment Papers that debunk common myths in the history of mathematics @MichaelGreinecker: certainly depends on the tone. you should have taken my original comment with a grain of salty joke :) I'm going to ask a question regarding the history of probability and measure theory on MSE in 5 minutes, may be of your interest. Dec 4 comment Papers that debunk common myths in the history of mathematics @MichaelGreinecker: oh no, have you just critisized Kolmogorov? Sep 24 awarded Autobiographer Aug 3 comment Sufficiency of stationary policy for negative stochastic dynamic programming @MichaelFanZhang: it may become unbounded ($-\infty$), but that's in fact important to negative dynamic programming. Also, since $g$ is sign-semidefinite, the sum is always negative so its expectation is well-defined, even though it may be $-\infty$ as well. Did I answer your question? Jul 22 comment Quotients of standard Borel spaces @Burak: I see your point, indeed some Borel selection would be needed. Jul 22 awarded Inquisitive Jul 21 comment Uniformization/measurable selection theorems I am also aware of the updated version of Wagner's paper published in 1979, where he claims to add more results know in the Russian literature by that time, however I do not have an access to that. Jul 21 asked Uniformization/measurable selection theorems Jul 20 revised Existence of an universally measurable pullback deleted 175 characters in body Jul 20 accepted Coupling of non-probability/sub-probability measures Jul 20 accepted Particular neighborhoods of analytical sets Jul 20 asked Stronger version of linearity for functions of measures Jul 20 comment Quotients of standard Borel spaces @Burak: for some reason I was not notified, so I just came across your comment. Do you mean here, that if $E$ is smooth then $X/E$ is at least analytic, and if $X/E$ is a standard Borel space, then $E$ is smooth? Also, isn't the surjectivity in the 2nd sense (meeting each $F$ class) equivalent to bireducibility by definition? Jul 20 comment Inverse of a Borel surjection I was exactly trying to use a Borel set which does not have a Borel uniformization, and relate it to a $\mathrm{Gr}(f)^{−1}$ via an isomorphism, however that did not work. Your idea of using a projection is really nice, thanks. Also, there is a survey of measurable selection theorems (Wagner 1977-79). Do you know if there was any progress after that, maybe another survey paper? Please tell me in case I should make it a separate question.