All Questions
Tagged with measure-theory integration
2,732
questions
4
votes
1
answer
55
views
If $g_1, g_2\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ are equal locally $\mu$-almost everywhere, then $T_{g_1}=T_{g_2}$.
Background
Suppose that $(X,\mathscr{A},\mu)$ is an arbitrary measure space, that $p$ satisfies $1\leq p<+\infty$, and that $q$ is defined by $\frac{1}{p}+\frac{1}{q}=1$. Let $g$ belong to $\...
- 2,493
0
votes
1
answer
46
views
If $A$ is Borel measurable, is $\int_0^21_{A}(x)\int_{x-4}^x\mathrm{d}u\mathrm{d}x=0$?
Consider the simple double integral $I(A)=\int_0^21_A(x)\int_{x-4}^x\mathrm{d}u\mathrm{d}x$ where $A$ is a set in the Borel $\sigma$-algebra over $\mathbb{R}$. I want to check the very simply question ...
- 362
0
votes
1
answer
40
views
Defining a measure on a finite dimensional Hilbert space
I am reading An Introduction to Infinite-Dimensional Analysis by Da Prato. Let $H$ be a $d$ dimensional Hilbert space where $d < \infty$ and $L^+(H)$ the set of symmetric, positive, and linear ...
- 6,277
0
votes
0
answers
13
views
Weak and strong integrability of a mapping $f : X \to E$ on a general measure space $X$ and locally convex $E$
Let $(X, \Sigma, \mu)$ be a measure space and $E$ be a locally convex toplogical vector space.
Let us consider a mapping $f : X \to E$. Then, we have two notions of integrability. That is, weak ...
- 7,829
4
votes
1
answer
544
views
Why are Lebesgue integrals defined as a supremum and not as a limit?
We may approximate a bounded nonnegative function $f$ by simple functions $\phi$. Then the standard way of defining the Lebesgue integral of $f$ is
$$ \int f d\mu := \sup \Bigg\{ \int \phi : \phi \...
- 6,277
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
0
votes
3
answers
88
views
Proving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$
Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$.
I would like to prove that
$$\int_D \int_D f(...
- 6,277
14
votes
4
answers
3k
views
Why Learn Measure Theory and Lebesgue Integration?
As someone who has taken two semesters of real analysis, having been exposed to the rigorous definition of the Riemann-Stieltjes integral - why should I learn Lebesgue integration? The Riemann ...
- 365
1
vote
1
answer
61
views
$e^{-\lVert x \rVert ^2}$ is integrable over $\mathbb{R}^n$ and $\int_{\mathbb{R}^n} e^{-\lVert x \rVert ^2} = \pi^{n/2}$
$e^{-\lVert x \rVert ^2}$ is lebesgue integrable over $\mathbb{R}^n$ and $\int_{\mathbb{R}^n} e^{-\lVert x \rVert ^2} = \pi^{n/2}$. I'm a bit lost on this, I've tried to use a particular result about ...
- 800
0
votes
0
answers
62
views
Showing the following function is nonnegative
Let $D \subset \mathbb{R}$ and $f: D \rightarrow \mathbb{R}$ $g: \mathbb{R} \times D \rightarrow \mathbb{R}$ be nonnegative functions. Let $\mu$ be a measure on $D$ such that $\mu(D) < \infty$. I ...
- 6,277
0
votes
2
answers
59
views
Need help understanding the switching of integral limits with Fubini's theorem: $\int_0^cm(\{x:|f(x)|>t\})dt=\int_{\{x:c\geq |f(x)|\geq 0\}}|f(x)|dx$
Let $f\in L^1(\mathbb{R}, m)$ with $m$ being the Lebesgue measure and $c > 0$. I have to admit that I am quite bad with using Fubini's theorem outside of calculus type problems of abstract proofs. ...
- 1,221
3
votes
1
answer
51
views
$\mathscr{L}^{p_1}(\mathbb{N},\mathscr{A},\mu)\subseteq\mathscr{L}^{p_2}(\mathbb{N},\mathscr{A},\mu)$: $1\leq p_1<p_2<+\infty$, $\mu$ counting measure
I need to prove the following:
Suppose that $1\leq p_1<p_2<+\infty$. Let $\mu$ is the counting measure on the $\sigma$-algebra $\mathscr{A}$ of all subsets of $\mathbb{N}$. Then $\mathscr{L}^{...
- 2,493
1
vote
0
answers
50
views
Integration with respect to finite Radon measure
Let $ u \in BV( \mathbb{R}^N). $ We know that
$$
\DeclareMathOperator{\Div}{div}
\DeclareMathOperator{\dm}{d\!}
\int_{\mathbb{R}^N} \left|Du\right| = \sup\left\{ \int_{\mathbb{R}^N} u\Div \varphi\dm ...
- 187
1
vote
0
answers
38
views
Layer cake representation and function with compact support
My question is related to the posts here and here, but my setup is slightly different.
For a real-valued random variable $X$, and a function $\varphi: \mathbb{R} \to \mathbb{R}$ that has support in ...
- 361
0
votes
1
answer
55
views
"Leibniz's rule" for $t\in\mathbb{R}^n$
I am looking for a reference giving a measure-theoretic proof of a claim from the German Wikipedia. I have searched the references given on that site, as well as the English speaking Wikipedia and all ...
- 1