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Doing math.


Jun
1
answered Proof of $\int\limits_{A}f=\int\limits_{\mathbb{R}}f{1}_A$ for the Lebesgue integral
Jun
1
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 $\int_{A} f=\int_{A}f+\int_{A^{c}}0=\int_{A}f1_{A}+\int_{A^{c}}f1_{A}=\int_{\mathbb{R}}f 1_{A}$.
May
31
revised Integrals and Dirac delta measures
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May
31
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 $\int X\,d\delta_{k}=\int X^{+}\,d\delta_{k}-\int X^{-}\,d\delta_{k}=X^{+}(k)-X^{-}(k)=X(k)$.
May
31
revised Integrals and Dirac delta measures
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May
31
comment Integrals and Dirac delta measures
Yeah, it looks correct now.
May
31
revised Integrals and Dirac delta measures
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May
31
answered Integrals and Dirac delta measures
May
31
comment Integrals and Dirac delta measures
Should the right-hand side be $X(0)$? Or at what point is that dirac measure standing on?
May
31
awarded  Citizen Patrol
May
31
comment Prove that $\sin(x+\frac{\pi}{n})$ converges uniformly to $\sin(x)$.
@Belgi: where in the definition of uniform convergence does $\delta$ appear?
May
31
comment Is the box topology good for anything?
Sure; and thanks for doing it. Such things which require correction are important to point out:)
May
31
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?
May
31
revised Is the box topology good for anything?
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May
31
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:)
May
31
revised Is the box topology good for anything?
Huge edit.
May
31
revised Is the box topology good for anything?
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May
31
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.
May
31
revised Is the box topology good for anything?
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May
31
answered Is the box topology good for anything?