Thomas E.
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 Jun1 comment Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral With this lemma it works. If you want to avoid working suprumems over sets in equalities, you may do something like the following too (you already got the idea). $"\Rightarrow"$: Let $0\leq s\leq f$ be arbitrary simple function, and define $s_{1}=s 1_{A}$ and $s_{2}=s 1_{A^{c}}$, whence $s=s_{1}+s_{2}$ and $s_{1}(x)\leq f(x)$ for $x\in A$ and $s_{2}(x)\leq f(x)$ for $x\in A^{c}$. Now using what you had already proven $\int_{R} s=\int_{R} s 1_{A}+s 1_{A^{c}}=\int_{R}s 1_{A}+\int_{R}s 1_{A^{c}}=\int_{A} s_{1}+\int_{A^{c}} s_{2}\leq \int_{A}f+\int_{A^{c}}f$. (Continues below) Jun1 comment Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral There was a tiny flaw, which I should've fixed, and I saw you posted your own:) I will read it soon. Jun1 answered Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral Jun1 comment Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral Have you proved that for any non-negative measurable function $f$ there exists a nondecreasing sequence of simple functions $(\psi_{i})$ so that $\psi_{i}\to f$ point-wise? With this and monotone convergence theorem you may conclude the result. Or you may want to notice that $\int_{A} f=\int_{A}f+\int_{A^{c}}0=\int_{A}f1_{A}+\int_{A^{c}}f1_{A}=\int_{\mathbb{R}}f 1_{A}$. May31 revised Integrals and Dirac delta measures deleted 9 characters in body May31 comment Integrals and Dirac delta measures @rosie: We can always do such a composition $X=X^{+}-X^{-}$ where $X^{+}=\max\{X,0\}$ and $X^{-}=-\min\{X,0\}$. Then by using the previous step (since both $X^{+}$ and $X^{-}$ are non-negative and measurable) we get $\int X\,d\delta_{k}=\int X^{+}\,d\delta_{k}-\int X^{-}\,d\delta_{k}=X^{+}(k)-X^{-}(k)=X(k)$. May31 revised Integrals and Dirac delta measures deleted 468 characters in body May31 comment Integrals and Dirac delta measures Yeah, it looks correct now. May31 revised Integrals and Dirac delta measures added 639 characters in body May31 answered Integrals and Dirac delta measures May31 comment Integrals and Dirac delta measures Should the right-hand side be $X(0)$? Or at what point is that dirac measure standing on? May31 awarded Citizen Patrol May31 comment Prove that $\sin(x+\frac{\pi}{n})$ converges uniformly to $\sin(x)$. @Belgi: where in the definition of uniform convergence does $\delta$ appear? May31 comment Is the box topology good for anything? Sure; and thanks for doing it. Such things which require correction are important to point out:) May31 comment Is the box topology good for anything? @SamL. Thanks for pointing that out. That identity would have not been mandatory but rather a shortcut: but clearly it is exactly as you pointed out. I edited it once again, maybe this should fix the problem? May31 revised Is the box topology good for anything? added 224 characters in body May31 comment Is the box topology good for anything? I edited the answer accordinly. It is fine as it is now too (although less practical), but I will also try to find some conditions where they coincide. But if nothing else, my answer is atleast an elaboration of dfeuer's comment as I promised:) May31 revised Is the box topology good for anything? Huge edit. May31 revised Is the box topology good for anything? deleted 453 characters in body May31 comment Is the box topology good for anything? @t.b.: good point, there might be a detail or an extra assumption that I have overlooked or forgotten. I will try to find this from somewhere. For now it seems that $\tau_{\sup}\subset$ Box topology.