T. Eskin
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 Dec 25 reviewed Reviewed integral of $\ln x$ from 0 to 1 Dec 25 revised Difference between $\lim_{n\to\infty}P(|X_n-X|<\epsilon)=1$ and $\lim_{n\to\infty}P(|X_n-X|=0)=1$ added 29 characters in body Dec 25 revised Difference between $\lim_{n\to\infty}P(|X_n-X|<\epsilon)=1$ and $\lim_{n\to\infty}P(|X_n-X|=0)=1$ added 29 characters in body Dec 25 answered Difference between $\lim_{n\to\infty}P(|X_n-X|<\epsilon)=1$ and $\lim_{n\to\infty}P(|X_n-X|=0)=1$ Dec 25 comment Proving $\lim_{n\to\infty}P\,(X_n=X) = 1$ when the sequence of random variables $X_n$ converges to $X$ a.s. This is not true. Take $X_{n}=\frac{1}{n}$ for all $n\in\mathbb{N}$. Then $X_{n}\to 0=:X$‚ but $P(X_{n}=X)=P(\frac{1}{n}=0)=0$ for all $n\in\mathbb{N}$. Hence $\lim_{n\to\infty}P(X_{n}=X)=0$. Dec 25 answered A question about the contractibility of the Sierpinski space Dec 24 answered Norm of vector in $\mathbb{R}^3$ with multiple Dec 24 answered Compact and open subspaces using products Dec 24 comment Compact and open subspaces using products It's true that $C\times K$ is a compact subspace of $X\times Y$, but taking $\{U\}$ as its open cover, the only finite subcover would be $\{U\}$ again. Dec 24 comment Sets of second category-topology Do you mean a countable union? Dec 22 comment Urysohn's metrization theorem and Borel image @Martin By $N_{2}$ I meant second countability and by $T_{3}$ regularity. And thanks a lot, your comments have been a valuable input to this question. Dec 22 revised Urysohn's metrization theorem and Borel image edited body Dec 22 revised Urysohn's metrization theorem and Borel image Rephrased the question a little. Dec 22 comment Urysohn's metrization theorem and Borel image @Martin. Thanks. I knew the connection of $G_{\delta}$ sets and completely metrizable sets, but it didn't occur to me to use it in this context. This is in fact a very nice a way of proving this. Do you think that $N_{2}$ and $T_{3}$ are alone enough for $f(X)$ to be Borel? I think I will slightly rephrase my question in this thread. Dec 22 revised Set of all finite subsets of $\mathbb{N}$ is a countable set Edited the word {N} in title and in the question. Dec 22 comment Urysohn's metrization theorem and Borel image @SamL. Very nice point that you're saying. It also answers one part of my question. Namely, that it does not follow from the topological properties as they stand, but explicitly from the choice of $f$. And the proof that I am looking at uses Urysohn's lemma countably many times appropriately with second countability of $X$, and does not give the explicit function. However, I am looking at some other results in measure theory that use this property quite freely. E.g. when embedding a Polish space to the Hilbert cube and assuming that $f(X)$ is a measurable set respect to a strict Borel measure. Dec 22 revised Urysohn's metrization theorem and Borel image deleted 7 characters in body Dec 21 asked Urysohn's metrization theorem and Borel image Dec 20 revised A continuity example Added LaTeX Dec 20 revised Set of measure zero and limsup of its covering edited LaTeX.