# Tag Info

## Hot answers tagged lebesgue-integral

9 votes
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### Can we conclude that $f=g$ a.e. if $\int _Efd\mu =\int _Egd\mu$ for all measurable sets $E$?

There are measure spaces where all measurable sets have measure $0$ or $\infty$. In such a space, let $f$ and $g$ be two different positive constants. [EDIT] More generally, suppose there is a ...
• 450k
4 votes
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### Existence of a sequence $\mu(I_j \cap I_k)= 0$ such that $\lim_{j\to \infty}\mu(I_j) = b-a$

$\mu(I_j \cap I_k)= 0$ for $j \ne k$ implies that $\sum_j \mu(I_j)=\mu(\bigcup_j I_j) <\infty$ and hence, $\mu (I_j) \to 0$.
• 37.3k
4 votes
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### Doubts in calculatating $\int_0^\infty \frac{n\sin x}{1+n^2x^2}\,\mathrm{d}x$

Yes, your estimate is correct, as $$0\leq(nx-1)^2=n^2x^2-2nx+1$$ implies that $$1+n^2x^2\geq2nx,$$ and the rest of the estimates are clear. Now the reason why you can ignore the point $x=0$ is because ...
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2 votes
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• 4,583
1 vote
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Suppose $y \in (a,b)$ and choose $y' \in (a,y)$. Then $F(y) = F(Y') + \int_{y'}^y F'(t)dt = \int 1_{[y',y]}(t) F'(t)dt$ Let $y_n \to b$ (with $y_n \in (a,b)$, of course), then since $|1_{[y',y_n]}(t) ... • 173k 1 vote ### Analysis of an expression involving a function on$\mathbb R^n$. Related to limits, supremums and translations. About your approach. The function$f$is bounded outside$B(0,1/2)$, i.e.,$|f(y)| < C$for some$C > 0$and all$y \not \in B(0,1/2)$. Note that for every small enough$\xi\$, the symmetric ...
• 4,909

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