# Questions tagged [lebesgue-integral]

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

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### Evaluating the limit of the Lebesgue integral $\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$

I am stuck on this integral $$\lim_{n\to\infty}\int_0^{2n\pi}\frac{(n+x)\sin(x/n)}{x(1+x)^2}\,dx$$ I have learned MCT and DCT, but I don't how they might be applicable. Any hint would be appreciated.
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### Does pointwise convergence imply measurability? and why?

$E$ is a Banach space and for every $u_0 \in E$ and $\{u_n\} \subset E$ with $u_n \to u_0$, we have $$S(t)u_n \to S(t)u_0$$ pointwise in $t \geq 0$. Why $t \mapsto S(t)u_0$ is measurable there ?
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### Question about applying Dominated Convergence Theorem

Question: $\phi_n(x)=\int_{x_0}^xf(t,\phi_n(t))dt$, where $\phi_n(x)$ is continuous on $[a,b]$ and $f$ is continuous and bounded on $[a,b]\times(-\infty,+\infty)$. If $\phi_n(x)$ converges uniformly ...
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### Clarifying weird integral notation

I would like your help to understand the following notation for integrals which I have never seen. Consider an integral $$\int_{a}^b 3 \text{ }d X$$ where $X$ is a random variable. What does this ...
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### Why does the monotone convergence fail here

I am looking at the example $f_{n}(x) = n \chi_{(0, \frac{1}{n}]}$. This converges to $0$ pointwise and graphing it out we can see that its a series of rectangles of area 1 but with growing height. I ...
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### If $f$ non-negative and bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}}f^{2} d \lambda < \infty$

I am trying to show if $f$ is a non-negative function that is bounded and $\int_{\mathbb{R}}f d \lambda < \infty \Rightarrow \int_{\mathbb{R}} f^{2} d \lambda < \infty$ Where d$\lambda$ denotes ...
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### $\int_\mathbb Z 3^{-|x|}d\zeta(x)$ [closed]

Let $\zeta$ be the counting measure on $(\mathbb Z,\mathscr P(\mathbb Z))$. Calculate $\int_\mathbb Z 3^{-|x|}d\zeta(x)$. How can I calculate this integral?
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### Intersection of meaning between Lebesque Measure and Natural Density

FYI: the articles linked and their content within are well-known in number theory, so number theory is a tag (correct if need be). $\textbf{Background}$: This article by Maier states that a given ...
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1 vote
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### Is this function continuous (when $U$ has no flat regions)?

Apologies for that useless modifier in the brackets in the title -I had to add that to avoid "duplicate titles". It is most natural that there will be multiple questions with the title "...
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1 vote
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