All Questions
55
questions
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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
3
votes
2
answers
53
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If $\int_Afd\mu\geq0$ for all $A\in\mathscr{A}$, then $\int f\chi_Ad\mu=0$ for $A=\{x\in X:f(x)<0\}$
I am self-studying measure theory using Measure Theory by Donald Cohn. I am confused by his proof of the following result:
Corollary 2.3.13$\quad$ Let $(X,\mathscr{A},\mu)$ be a measure space, and ...
- 2,493
3
votes
1
answer
58
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Question About Function Integrability - Proposition 2.3.10 from Measury Theory by Donald Cohn
I am self-studying measure theory using Measure Theory by Donald Cohn. The book makes the following definition:
Definition$\quad$ Suppose that $f:X\to[-\infty,+\infty]$ is $\mathscr{A}$-measurable ...
- 2,493
2
votes
0
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66
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Question on the Construction of the Integral in Measure Theory
I am self-studying measure theory, and I got some trouble understanding the construction of the integral. Here is the first two stages of the construction:
Stage 1$\quad$ We begin with the simple ...
- 2,493
0
votes
1
answer
48
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Nice application of dominated convergence theorem
Let $\delta \in \mathbb{R}$, $$f(x)=\frac{sin(x^2)}{x}+\frac{\delta x}{1+x}.$$
Show that $$\operatorname{lim_{n\to \infty}} \int_{0}^{a}f(nx)=a\delta$$ for each $a>0.$
I am unable to find ...
- 645
1
vote
1
answer
86
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Integral over some measure is the same as integrating over the measure of the preimages of some intervals
Sorry for the somewhat confusing question. I'm currently working on this problem:
Let ($\Omega, \mathcal{A}, \mu$) be a σ-finite measure space and $f: \Omega \rightarrow \overline{\mathbb{R}}$ with $\...
- 718
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Integrability of a function $g$ satisfying $\mu(A)=\int gd\nu$
Fix a measure space $(S,\Sigma,\mu)$ and let $(E_n, n\geq 1)$ be a sequence of disjoint measurable sets so that $\mu(E_n)<\infty$ for all $n$. Now fix another measure $$\nu(A)=\sum_{n=1}^{\infty} 2^...
- 362
0
votes
1
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109
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$\lim_{n\to\infty}\int_X f_n d\mu=2023$ and measures
Let $(X,\mathcal{A},\mu)$ be a measure space and let $\{f_n\}_{n\geq1}$ be a sequence in $\mathcal{M}_{\bar{\mathbb{R}}}^+(\mathcal{A})$. Assume that $f_n\to f$ for $n\to \infty$ and that
$$\lim_{n\to\...
user1197800
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Joint measurability of discrete integral.
I'm currently considering the following map: Let $\mathcal{F}$ be a subset of some $L^2$ space with respect to a probability measure on $\mathbb{R}$ and let $X=(X_1,X_2,\ldots,X_n)$ be points on the ...
4
votes
3
answers
311
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On the definition of locally integrable functions on an abstract measure space.
Recently I've been able to find a very cheap copy of the nice monograph [1], where I can find (chapter 10, §1, p. 163) the following general definition of Locally integrable function (respect to a ...
- 10.1k
2
votes
1
answer
84
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Does this condition about two measures $p$ and $q$ imply existence of $p$-integrable function that is not $q$-integrable?
Context :
I am trying to characterize some properties of barycenters of measures on a probability space, the following question arises. More precisely I want to restrict the support of a measure on ...
- 6,076
1
vote
1
answer
118
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Monotone convergence theorem on a function of two variables (integrating over one variable)
Consider a measurable space $\big( E \times F, \mathcal{E} \otimes \mathcal{F} \big)$, an $(\mathcal{E} \otimes \mathcal{F})$-measurable and positive function $f$, and a measure $\nu$ on $\big( F, \...
- 385
2
votes
2
answers
253
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Why is this theorem Cavalieri's Principle?
I've been studying measure Theory, using Bartle and Folland, mostly Bartle. Just this morning I've come in contact with the layer cake representation of a function while I was trying to understand ...
- 161
2
votes
2
answers
80
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Is this sufficient to conclude $L^{1}$ convergence?
Let $f_{n}:[-1,1]\to \Bbb{R}$ be a sequence of Measurable functions such that $|f_{n}|\leq\frac{1}{x^{4}}$ and $\displaystyle\int_{[-1,1]}(f_{n}(x))^{4}x^{2}\,dx\leq 1$ and $f_{n}$ converges in ...
- 1,285
2
votes
1
answer
383
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Functions that are integrable with respect to all probability measures [duplicate]
I am interested to know which measurable functions are integrable with respect to all probability measures, i.e., all $f$, for which:
$$ \int \| f \| ~{\rm d}\mathbb{P} < \infty,$$
where $\mathbb{...
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