Thomas E.
Reputation
4,761
Next privilege 5,000 Rep.
Approve tag wiki edits
 Aug 13 comment Baire Category Theorem By the way, this works also for non-separable spaces. And you could try to show the complement of this statement which says that a countable intersection of open dense sets is dense. It's quite simple, too. You begin with an arbitrary open ball and start choosing closed balls inside it inductively so that the radius goes to zero and the $n$:th step ball is a subset of $n$ first members of the dense open sets. The intersection of these closed balls is a singleton by completeness: it belongs to the intersection of the open dense sets and our original ball, proving the claim. Aug 13 comment $L^p$-norm of a non-negative measurable function Hint: Assume first that $f$ is a simple function and show that the equation holds. Then for your non-negative measurable $f$ choose a nondecreasing sequence of simple functions converging point-wise to $f$ and use monotone convergence theorem. Aug 13 comment Outer measure of a union of 2 subsets of disjoint measurable sets of real numbers. Why are subsets of measurable sets bounded? Consider for example $]0,\infty[\subset \mathbb{R}$, where $\mathbb{R}$ is measurable and $]0,\infty[$ is unbounded. Aug 6 comment How to bound' $L^\infty$ by the constant function $1$ @t.b. Thanks. I overlooked that part and it makes perfect sense now. Aug 6 comment How to bound' $L^\infty$ by the constant function $1$ But this bound only works almost everywhere. Sure, you can find a representative of $f$ from same equivalence class for which this works, but in general $f$ can be unbounded. Right? Aug 1 awarded Benefactor Aug 1 accepted Disintegration theorem, a reference needed Aug 1 comment Disintegration theorem, a reference needed @ByronSchmuland: Thanks, I will also look that up :-) Aug 1 comment Disintegration theorem, a reference needed @ByronSchmuland: I'm looking for something covering the existence of disintegrations in atleast as general setting as presented in Wikipedia, which is Radon spaces. Particularly I would be prefer to avoid probabilistic conditioning approaches and keep it on a general level. I think Fremlin's book was a perfect match for this search, but in any case, I'm grateful for every answer made in this topic. Aug 1 comment Disintegration theorem, a reference needed So far Fremlin's book has been the most impressive and it is exactly what I was looking for. Not only is this opus amazing but so are his other books of the same 'series'. I also looked up rest of your suggestions and they were useful too, thank you. The bounty can be awarded after 1 hour, so until then if nothing better (which I doubt) appears then it surely belongs to this answer. Jul 31 comment Disintegration theorem, a reference needed @EdGorcenski. You ignored the second part of the sentence which gave some specification on its style. The book in question is (most likely, judging from its appearance) written with a typewriter machine. Jul 31 comment Disintegration theorem, a reference needed Since none of the (so far) provided books have met my interests and this is a rather important matter for me, I have raised a 50 rep bounty for this question. Jul 31 awarded Promoter Jul 31 comment The metrizable space may be not locally compact @Paul. Since in metric spaces compactness is equivalent with sequential compactness then it suffices. Take any open ball, it contains Cauchy sequences that have a limit in $\mathbb{R}$ but not in $\mathbb{Q}$ (a sequence that converges to an irrational number), then by closuring this open ball in $\mathbb{Q}$ you at most obtain more rationals in the set and not these limits a priori. Thus the closure of every open ball has sequences with no convergent subsequences and is hence non-compact. Jul 31 comment The metrizable space may be not locally compact You have sequences that lie entirely in $\mathbb{Q}$ but converge to irrationals in $\mathbb{R}$. By taking a closure of a set in the relative topology of $\mathbb{Q}$ you at most gain more rationals to the set, not these limits for example. For this reason, from the closure of every open ball in $\mathbb{Q}$ you find sequences with no convergent subsequences. Jul 31 comment Disintegration theorem, a reference needed @did: Yes, I have that book in my shelf and I'm reading it as we speak. However, I couldn't find anything related to what you suggest. My best guess would be that this topic could be covered under Chapters 6.33-34, but it looks like they aren't. Maybe he covers these topics with different terminology, but I highly doubt it would be dealt before the Radon-Nikodym section. Altogether, so far I haven't been successful in finding this book useful in this particular context. Jul 30 comment In mathematics, what is meant by induction? The Wikipedia article on this topic can be useful to read: en.wikipedia.org/wiki/Mathematical_induction Jul 28 comment Disintegration theorem, a reference needed @StefanHansen: Could be, I'm not that familiar with conditional probability theory. If you can point some sources then I would be glad to go them through. And for others: I'm also looking for sources that deal with general metric spaces (locally compact Polish, compact?) with the push-forward framework and disintegration. Mainly in non probabilistic setting. Jul 27 asked Disintegration theorem, a reference needed Jul 23 comment If a subset of $\mathbb{R}$ is closed and bounded with respect to a metric equivalent to the Euclidean metric, must it be compact? Just an idea: what if $\hat{d}$ would be such that every bounded set would also be totally bounded respect to this metric (as in the Euclidean case). Would this be sufficient for these statements to hold? And would it also be a necessary condition?