# 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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### A question on Lebesgue integral

Let $f\in L^1(\mathbb{R}^n)$. I want to prove that when $m(B-B’)\to 0$ $$\left|\int_{B’}f(x)dx-\int_{B}f(x)dx\right|\to 0.$$ In Riemann integral, if $f$ is absolutely integrable on $[a,b]$, then $|f|$...
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### Substitution/Transformation to integrate $(1+\langle x,x\rangle)^{-2}$ over $\mathbb{R}^2$

I want to integrate $f:=(1+\langle x,x\rangle)^{-2}$ over $\mathbb{R}^2$. Geometrically this problem is finding the volume of the stack of disks $$\frac{1}{\sqrt{z}}-1\ge x^2+y^2$$ I'm sure there is ...
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### Find Lebesgue Integral: $\lim_{n\rightarrow\infty}\int_0^2f_n(x)dx$

Consider a sequence of functions $f_n:[0,2]\rightarrow\mathbb{R}$ such that $f(0)=0$ and $f(x)=\frac{\sin(x^n)}{x^n}$ for all other $x$. Find $\lim_{n\rightarrow\infty}\int_0^2f_n(x)dx$. My ...
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### Bochner integral in a direct sum of Banach spaces

Let $\mathcal{B} = \mathcal{B}_1\oplus\ldots\oplus \mathcal{B}_n$ be a direct sum of Banach spaces $\mathcal{B}_i$ each with norm $\|\cdot\|_{\mathcal{B}_i}$. The Banach space $\mathcal{B}$ has many ...
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### Notation on Measure Theory and Integration

What is the meaning of $L^{2}_{loc}(\mathbb{R}, L^{2}(\Omega))$?, where $\Omega\subset\mathbb{R}^{d}, d\geq 1; L^{2}(\Omega)\equiv \mathcal{L}^{2}(\Omega)/ker(\|\cdot\|_{2})$ Is there some good ...
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### I need to evaluate $\iint_S xdydz + (x+y)dz dx+(x^2+2z)dxdy$

My problem asks me to evaluate the integral (using direct integration) $$\iint_S (x)dy\wedge dz + (x+y)dz\wedge dx+(x^2+2z)dx\wedge dy$$ Being $S$ the surface of the solid limited by the following ...
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### If $f$ is Lebesgue integrable, then show that $h(u)=\int_{\mathbb{R}} e^{iux} f(x) dx$ , for real $u$.

Suppose we have a function $h$ which is continuous and Lebesgue integrable on $\mathbb{R}$. We have $f(x)=\frac{1}{2 \pi} \int_{\mathbb{R}} e^{-iux} h(u) du$, for all real $x$. If $f$ is Lebesgue ...
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### Lebesgue's fundamental theorem of calculus

The second part of the fundamental theorem of calculus states that If $f:[a,b]\to\mathbb{R}$ is a Lebesgue-Integrable function and $F$ is a primitive of $f$, then $\int_{a}^{b}f(x)dx=F(b)-F(a)$. I ...
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### Lp space Example

How are spaces connected $L_{\infty}(E)$ and $L_{p}(E)$, $|E| = \infty$? $$f \in L_\infty, \text{ but } f\notin L_1 \quad f = \frac{1}{x} \quad E = [1, \infty)$$ What can we say about reverse ...
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### Integrability of $\frac{1}{(x^2+y^2+z^2)^a}$ on $E=\{(x,y,z)\in \mathbb{R}^3: z>1, \ z^2(x^2+y^2)<1 \}$

Let $E=\{(x,y,z)\in \mathbb{R}^3: z>1, \ z^2(x^2+y^2)<1 \}$ and $$f_{a}(x)=\frac{1}{(x^2+y^2+z^2)^a}$$ I need to find all $a\in \mathbb{R}$ such that $f_a\in L^1(E).$ I already know a ...
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### Help to evaluate the integral $\iint_D\frac{y}{\sqrt{x^2+y^2}}dxdy$
I'm solving a problem about integrals in curves, and I got this integral: $$\int_1^2\int_1^2\frac{y}{\sqrt{x^2+y^2}}dxdy.$$ I have been struggling to solve it. I'm sure i have to do some variable ...