# 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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### Is there a solution manual for Royden fourth edition?

I bought the fourth edition of Royden Real Analysis, this book is awesome and is quite different of third edition that has less excersices. I have the solution manual for the third edition. Is there ...
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### limsup of continuous function is measurable.

I want to show that if $F$ is continuous on $[a,b]$ then $$\limsup_{h \rightarrow0, h>0} \frac{F(x+h)-F(x)}{h}$$ is measurable. By the definition of $\limsup$ we can write \begin{align} &\...
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### Prove that $\lim_{n \to \infty} \int_{[0,1]}{x^n}\, dx = 0$. Where $\int$ represents Lebesgue integration.

Please check my proof, thank you. Prove \begin{align*} \lim_{n \to \infty} \int_{[0,1]}{x^n}\, dx = 0 \end{align*} Proof. Let $f_n(x) = x^n$. Since $f_n$ is a polynomial it is continuous and ...