All Questions
Tagged with measure-theory integration
2,732
questions
0
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1
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21
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Measurability of a family of parametric integrals assuming measurability of the integrand w.r.t. the parameter
Let $D$ and $E$ be measurable subsets of $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and $v : (x,t) \in D \times E \mapsto v(x,t) \in \mathbb{C}$.
Assume that the maps $v(\cdot, t)$, $t \in E$ ...
- 5,849
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0
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28
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Strict inequality of functions only allows to deduce a non-strict inequality of the expected value of said function
In a proof of Jensen's Inequality that I am reading, the following is used:
If for a real valued random variable $X$, we have $X(\omega)<\beta$, then $\mathbb{E}[X]\leq \beta$.
Why can we deduce ...
- 365
1
vote
0
answers
68
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What's the definition of a line integral on a possibly disconnected curve?
I'm trying to understand this paper, and I see this integral (page 2):
$$
\int_{B\ \cap\ \mathcal{C}} (1 - y)dy,
$$
where $\mathcal{C}$ is the curve given by $x = ye^{1-y}$ and $B$ can be any ...
- 1,961
1
vote
1
answer
35
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Prove: If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.
I need to prove the following result:
Let $(X,\mathscr{A},\mu)$ be a measure space. If $f_1$ and $f_2$ are integrable, then $f_1\vee f_2$ is integrable over each $A\in\mathscr{A}$.
Here is my ...
- 2,493
0
votes
0
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32
views
Is there a simple proof that the Laplace transform of a signed measure $\mu$ on $\mathbb R$ uniquely determines $\mu$? [duplicate]
A Laplace transform of a signed measure $\mu$ on $(\mathbb R, \mathcal B(\mathbb R))$ is defined by
$$ f_\mu(s) = \int_{\mathbb R} e^{-\lambda t} d \mu(t), \qquad \forall s \in \mathbb R. $$
I know ...
1
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0
answers
58
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Prove: $F:M(X,\mathscr{A},\mathbb{R})\to\mathbb{R}$ defined by $F(\mu)=\int fd\mu$ is a linear functional. [closed]
I need to prove the following result:
Remark 4.30$\quad$ Let $M(X,\mathscr{A},\mathbb{R})$ be the collection of all finite signed measures on $(X,\mathscr{A})$. Let $B(X,\mathscr{A},\mathbb{R})$ be ...
- 2,493
0
votes
1
answer
39
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Integration with spherically symmetric measure in $\mathbb R^d$
Let $\mu$ be a finite spherically symmetric measure over $\mathbb R^d$, so that $\mu(TB) = \mu(B)$ for all orthogonal transformations $T: \mathbb R^d \rightarrow \mathbb R^d$ and Borel set $B$. Let $g:...
- 603
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64
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Find the limit $\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx$
Compute the limit$$\lim_{k \rightarrow \infty } \int_0^k x^{-k} e^{x^2/k^2}\sin(x/k)~\mathrm dx.$$
Completely stuck with this one. Some convergence theorem is obviously needed, but can't figure out ...
- 17
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votes
1
answer
59
views
Estimating integrals and measures over Hilbert space using finite dimensional projections
Let $H$ be a separable Hilbert space with orthonormal basis $\{e_k\}_{k \in \mathbb{N}}$. Let $P_n$ be the projection onto the $n$-dimensional subspace of $H$:
$$P_n x = \sum_{i=1}^n \langle x, e_i\...
- 6,277
0
votes
0
answers
39
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Integral over a sphere in $R^n$
Let $\sigma$ be the Lebesgue measure on the unit sphere $\mathbb S^{n-1}$ of $\mathbb R^n$.
Let $\Sigma$ be a semi-definite positive symmetric matrix in dimension $n$.
Is it possible to get a closed-...
- 369
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0
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62
views
Proof that the volume function is $\sigma$-additive
A $\textbf{box}$ $Q\subset \mathbb{R}^n$ is defined as $Q:= (a_1,b_1) \times \ldots \times (a_n,b_n)$ and the same with closed or half opened intervals where $a_i < b_i \in \mathbb{R}$ and in case ...
- 325
1
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91
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Differentiability of an integral depending on a parameter
Let $f$ be a function in $L^1 ([0,1])$, and for $y \in [0,1]$ consider $$F(y) = \int_{[0,1]} (1+ |f|)^y dx$$
Is $F$ differentiable in $(0,1)$? If it is, what is its derivative?
I know that that, given ...
- 11
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96
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Example of Hilbertian norm on the space of radon measures
Assume we are given compact subset $X \subset \mathbb{R}^n$. Consider the space of radon measures $M(X,\mathbb{R})$. I'm trying to find some Hilbert norm on this space either globally or locally. I ...
- 53
1
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0
answers
50
views
Parameter dependent functions integral
I got this question on a measure theory exam today, and after hours of discussing with my colleagues, im still quite confused. I have been able to prove the first point, but I am having trouble with ...
- 23
2
votes
1
answer
41
views
specific confusion on average integral
any idea why this is true? I am not able to figure out.
Given $f \in W^{1,p}(B(x, R))$, we want to prove that
$$
\left| \frac{1}{|B_{2^{-l}}(x)|} \int_{B_{2^{-l}}(x)} f - \frac{1}{|B_{2^{-l-1}}(x)|} \...
- 427
4
votes
1
answer
55
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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
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1
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46
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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
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1
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40
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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
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0
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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
$$ \int f d\mu := \sup \Bigg\{ \int \phi : \phi \...
- 6,277
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
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3
answers
88
views
Proving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$
Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$.
I would like to prove that
$$\int_D \int_D f(...
- 6,277
14
votes
4
answers
3k
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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
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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
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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