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Aug
24
comment the function $1/|x|^α$ in $R^n$
If $f:\mathbb{R}^{n}\to\mathbb{R}$, then what does $f(r)$ mean for $r\in[0,\infty[$?
Aug
23
answered Question about measurable sets $E_n$ such that $\lim_{n\rightarrow \infty}L^N(E_n) = 0$.
Aug
17
comment 2 jars with 50 balls each. Pick any ball with $>50\%$ probability.
@sTEAK. If you take one from each jar then the probability of getting a red is $1$.
Aug
16
comment Is it possible to have $D=\Bbb P$
@t.b. Yeah, the countability is important :) Baire implies that any countable completely metrizable topological space has an isolated point. If not, you could write the whole space as a countable union of nowhere dense closed sets (singletons), making the whole space to have an empty interior, which is a contradiction.
Aug
16
comment Is it possible to have $D=\Bbb P$
@t.b. Yeah, I meant an isolated point. And true, that also does the job. Thanks for providing an alternative way of seeing it.
Aug
16
revised Is it possible to have $D=\Bbb P$
added 462 characters in body
Aug
16
answered Is it possible to have $D=\Bbb P$
Aug
14
comment Is the set of irrationals separable as a subspace of the real line?
@CameronBuie. To add on Asaf's comment, it's worth noting that the word 'metric space' plays a key role there. Metrizability is hereditary and so is second countability. Since in a metric space separability (also Lindelöf) is equivalent with second countability, then every subspace of a separable metric space is separable (and Lindelöf, for that matter).
Aug
13
comment Baire Category Theorem
@AsafKaragila. True. The ZF part was so well hidden in the link that I totally missed it.
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.