# 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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### Prove $\lim_{n \rightarrow \infty} f(x) f(2^2x) f(3^2x) \cdots f(n^2x) = 0$ for $f: \mathbb{R} \rightarrow \mathbb{R}$ in $L^1(\mathbb{R})$.

Here's another question that I'm stuck on from my studies for an upcoming exam. This one comes from another practice preliminary exam. Problem Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a ...
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### Pass to the limit under the sign of the integral of $f_{n}(x)$ [closed]

I'm trying to investigate following limit: $$\lim_{n \to \infty}\int_{0}^{\frac{\pi}{2}}\frac{cos^{n}x}{1 + x^{3}}dx$$ I have a couple of questions 1. Is it possible to find the limit by the ...
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### Give an example function $f \in L_p(\mathbb {R}) \forall \: p > 7,$ but $f \notin L_p(\mathbb {R}) \: p \leq 7$ [closed]
Give an example function $f \in L_p(\mathbb {R}) \forall \: p > 7,$ but $f \notin L_p(\mathbb {R}) \: p \leq 7$