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.
7,939 questions
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Weak derivative in $L^1$-sense
In Wheeden and Zygmund's "Measure and Integration", 2nd edition, while talking about the differentiability properties of Fourier Transform, they mention the following:
Let $f$ in $L^1(\...
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If the integral of a real valued measurable function is smaller than infinity does that imply that the function is smaller than infinity a.e.?
Let $(X, \mathcal{A}, \mu)$ be a measure space and $f:X\rightarrow \mathbb{\bar{R}}$. Does $\int f d\mu<\infty$ imply $f<\infty$ $\mu$-a.e.?
EDIT: So I am trying to prove the contrapositive ...
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Justifying an estimate in the proof of $\lim_{t\to0}(f*\psi_{t})(x)=f(x)\int_{\mathbb{R}^{n}}\psi dm_{n}$ for $\psi \in L^{1},f \in C^{0}$
At the moment I am taking a remote course In Harmonic Analysis, and we started looking in to convolutions. In the lecture we defined:
Definition. the convolution of $f,g$ is
$$(f\ast g)(x) := \int_{\...
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How to show there exists a polynomial $p_j$ of degree at most $j$ such that $\int^b_a (u-p_j)^{(m)}=0\quad\forall\,m\leq j$, for $u\in W^{k,p}(a,b)$?
Question. $k\geq 1$ an integer, $p\in [0,+\infty]$, and $(a,b)$ a bounded open interval. Show by induction that for any $u$ in the Sobolev space $W^{k,p}(a,b)$, and for any non-negative integer $j\leq ...
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When can I replace functions that are equal almost everywhere by a specific representative?
Let $(X, \mathcal{F}, \mu)$ and $(Y, \mathcal{G}, \nu)$ be measure spaces. Consider the class of functions $[f]:X\times Y\rightarrow \mathbb{R}$ which are for each fixed $y\in Y$ measurable in $X$ (...
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Maximality of $\mathcal{L}^1$
I'm stuck in question 24 of Carlos Isnard's book "Introdução à Medida e Integração" (Introduction to Measure and Integration) from chapter 8 "Lebesgue and Borel $\sigma$-algebras".
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Finite Element Method for Laplace equation SPD
Problem setup: We want to show that the matrix obtained from Finite Element Method for the Laplace equation is symmetric positive definite.
Progress: The original pde is $\nabla^2 u = f$ on $\Omega$ ...
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Sequence of Riemann integrable functions which do not converge to a Riemann integrable functions
I am revising Measure theory and the reasons why we have to introduce an alternative definition of integral (which is Lebesgue integral). One of the main reasons why the Riemann integral “is not good ...
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Evaluating $\int_0^\infty \frac{\sin x}{x}\text{d}x$ using Fubini's theorem?
I found the following proof of evaluating the Dirichlet integral on ProofWiki (https://proofwiki.org/wiki/Dirichlet_Integral/Proof_1):
By Fubini's theorem
$$\int_0^\infty\left(\int_0^\infty e^{-xy}\...
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How to show $\{\upsilon\in H^{1}(0,1):\int_{0}^{1}\upsilon\,\mathrm{d}x=0\}$ is closed in $H^{1}(0,1)$?
Notation. $H^1(0,1)$ is the Sobolev space $W^{1,2}(0,1)$.
Question. How to show $V:=\{\upsilon\in H^{1}(0,1):\int_{0}^{1}\upsilon\,\mathrm{d}x=0\}$ is closed in $H^{1}(0,1)$?
My Attempt. Let $(\...
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Does $f<g$ $\mu$-a.e. imply $\int f d\mu <\int g d\mu$?
Let $(X, \mathcal{A}, \mu)$ be a measure space. Then it is true that $f\leq g$ $\mu$-a.e. implies $\int f d\mu \leq \int g d\mu$.
Is it true that $f<g$ $\mu$-a.e. implies $\int f d\mu < \int g d\...
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Continuous inclusion of mixed lebesgue spaces into lebesgue spaces
Consider some measurable function $f:\mathbb R^m\times\mathbb R^n\to\mathbb C$. Given $1\le p,q\le\infty$, we define the mixed norm
$$
\|f\|_{L^q(\mathbb R^n:L^p(\mathbb R^m))}=\left(\int_{\mathbb R^n}...
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Generalizing Riemann Integral for a General Measure
The following is question 9 of Carlos Isnard's book "Introdução à Medida e Integração" (Introduction to Measure and Integration) from chapter 6 "The Lebesgue and Riemann Integrals"....
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integrating over two non-negative functions that may take the value infinity
Let $(X, \mathcal{F}, \mu)$ be a measure space and consider non-negative extended real valued measurable functions $f, g:X\rightarrow [0, \infty]$.
Consider $$\int_X f(x)d\mu(x)-\int_X g(x)d\mu(x).$$
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Inductive proof of the change of variables theorem
I wanted to drop here an inductive proof of the Change of Variables Theorem (in integration), which might or might not be novel. If I'm not mistaken, MathSE serves as a repository, so self-answered ...
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