# 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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### How to show the following function is not Lebesgue integrable? [closed]

How to show that $\int_{||x||\ge 1} \frac{1}{||x||^n} dx$ where $x\in \mathbb{R}^n$ is not finite , i.e. $\frac{1}{||x||^n}$ is not integrable outside a ball of radius $1$ centered at $0$ ? I ...
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### Integration by parts formula with Lebesgue Integral and distribution function

I'm struggling to find a solution for the following problem: Let $f$ be an absolutely continuous function on $[a,b]$, let $\mu$ be a bounded Borel measure on $[a,b]$, and let $\Phi_\mu(t)=\mu([a,t))$ ...
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### Proving that the function $f(x) := |x|^{\frac{\lambda - n}{p}} (1- \psi(x))$ satisfies two specific properties related with limits and supremums.

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$, where $n \in \mathbb N$ is an arbitrary fixed integer that stands for the dimension of the euclidian space $\mathbb R^n$. In everything ...
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### Product of two Lebesgue measurable set is measurable

Let $A,B \subset \mathbb{R}$ two bounded and Lebesgue measurable sets. I have to show that $A\times B \subset \mathbb{R}^2$ is measurable and \begin{align*} \lambda(A \times B) = \lambda (A) \cdot \...
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### How does the often used picture about Lebesgue integration relate to the actual definition

Often when comparing Riemann and Lebesgue integration pictures like the following are used (credits to this Reddit post by user u/Analemmath) However the actual most common definition of Lebesgue ...
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### Stein Analysis 3.5.2 - kernels with average zero

Consider the problem given below from Stein and Shakarchi's Real analysis. Here is the same question being asked before, but my attempt is different, so I wanted to ask as a new post. I want to know ...