# Tagged Questions

For question about integration, where the theory is based on measures. So it's almost always used together with the tag [measure-theory], and its aim is to specify questions about integral, not only properties of the measure.

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### How to start an eigenvalue problem

I am stuck on this problem : This is an eigenvalue problem $$\phi''+ \lambda^2 x(x+2)^2 \phi =0\\\phi(1)=0\\ \phi(0)=0$$ I forget this kind of problems... please give me a hint or a clue ,cause I ...
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### Frullani's theorem in complex context, other examples

One has as application of Frullani's theorem in complex context that $$\int_0^\infty \frac{e^{-x\log 2}-e^{-xb}}{x}dx=\mathcal{Log} \left( \frac{1}{2\log 2}+i\frac{B}{\log 2} \right)$$ where I taken ...
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### $L^p \subset L^q$

Let $(X,M,\mu)$ be a measure space. Let $\Omega \subset X$ be a measurable set. We have $L^2(\Omega) \subset L^1(\Omega)$ . Can we have that $\mu(\Omega)< \infty$ ?
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### When we can permute between the integral and convex hull?

Is there a relation between the following expressions? $$\operatorname{conv}\left(\int_{0}^{t} f(s,x)ds :x \in A \right)$$ and $$\int_{0}^{t} \operatorname{conv}(f(s,x):x \in A)\ ds$$ where $A$ ...
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### Why is Lebesgue-Stieltjes a generalization of Riemann-Stieltjes? Moreover, is there an example where Lebesgue-Stieltjes is useful

I certainly have a question, but i don't know what the best title should be. Please edit the title if there is a better one :) And I believe, to get a better answer, it would be good to explain ...
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### Evaluation of $\int_{[0,\infty)}\biggl(\int_{[0,\infty)}2x\sqrt{y}e^{-x^2\sqrt{y}-y}dy\biggl)dx$.

Find the value of $$\int_{[0,\infty)}\biggl(\int_{[0,\infty)}2x\sqrt{y}e^{-x^2\sqrt{y}-y}dy\biggl)dx$$ My attept: Let $f(x,y)=2x\sqrt{y}e^{-x^2\sqrt{y}-y}$ to apply Tonelli theorem I did the ...
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### Prove that the Lebesgue integral of $f\chi$ equals to $0$ indicates that $f=0$ a.e.

Suppose $f: [0,1] \to \mathbb{R}$ is bounded, measurable, and $$\int_{[0,1]}f \chi_{[0,a)}\, d\mu = 0$$ for all $a \in [0,1]$. Prove that $f=0$ a.e. I know that if $\int_{[0,1]}f\, d\mu = 0$,...