All Questions
Tagged with measure-theory borel-measures
450
questions
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Can there be a Borel measurable bijection between $\mathbb{R}$ and $\mathbb{R}^2$ [closed]
I know a bijection exists between those sets, that it can be constructed quite explicitly and that it can't be continuous. I'm just not sure about the Borel measurability part.
0
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0
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17
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How to find a measurable, Lebesgue invariant bijection between the interval and the 3-dimensional sphere [closed]
From a book I was reading I have the following statement:
"It is a general fact of measure theory that there is a bijection
$f : [0, 1) → \mathbb{S}^2$ such that both $f$ and $f^{-1}$ take ...
1
vote
1
answer
70
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Why does $C=\{x\in\mathbb{R}:\nu(\{x\})\neq0\}$ belong to $\mathscr{B}(\mathbb{R})$?
Let $\nu$ be a finite measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Let $C=\{x\in\mathbb{R}:\nu(\{x\})\neq0\}$. I want to show that $\nu$ can be decomposed into the sum of a discrete meausre, a ...
- 2,493
1
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1
answer
33
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Compactly supported Borel measure $\mu$ on $\mathbb{R}^d$ [closed]
Definition :
If $\mu$ is any Borel measure on $\mathbb{R}^d$ then its support is defined as $$\text{supp}(\mu):=\{x\in\mathbb{R}^d:\text{every open neighbourhood of }x\text{ has positive measure}\}. $$...
- 371
3
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answers
47
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Is the Evaluation Map on Bounded Borel Measurable Functions Borel Measurable?
I am working with a set $I$, defined as the closed interval $[0,1]$, and a set $X$, which consists of all bounded Borel measurable functions defined on $[0,1]$. The uniform metric on $X$ is defined by:...
- 51
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0
answers
65
views
Borel $\sigma$-field and the product $\sigma$-field
I have a question about the proof of lemma 1.5 of the textbook Measure Theory, Probability, and Stochastic Processes by Le Gall.
Lemma 1.5: Suppose that $E$ and $F$ are separable metric spaces, and ...
- 41
2
votes
1
answer
32
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Finding measurable subsets of any given value?
I'm not sure if the following is true, but I would hope it is with the regularity properties of $\mu$.
Let $X$ be a locally compact Hausdorff space with $\mu$ a nonzero Radon measure on $X$. Then ...
- 1,399
-1
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1
answer
23
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Image of measurable sets under one to one (a.e) functions
I want to know about this question that is image of a measurable set under a one to one (almost everywhere) function, measurable?
Consider the following:
Let $(X, \mathcal{B}_X)$ and $(Y, \mathcal{B}...
- 1,053
0
votes
0
answers
38
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Let $\omega =\rho d\theta$ be a volue form of the circle $S^1$. Who are the diffeomorphisms of the circle that let $\omega$ invariant?
Consider the parametrization $\phi:]0,2\pi[\to S^1$ given by $\phi(\theta)=e^{i\theta}$. So the Lebesgue measure is given in local coordinates by the form $d\theta_z(\partial_z)=1$. I know that the ...
- 2,155
1
vote
0
answers
27
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Measure on the $d$-dimensional torus
I am looking for references, measure-integration theory where the $d$-dimensional torus $\mathbb{T}^d$ is treared rigorously: borel $\sigma$-algebra, measure functions, measures on $(\mathbb{T}^d,\...
- 616
1
vote
1
answer
68
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Existence of probability measures $P$ such that for any $x\in \Omega$ and closed set $x\not\in A$, the inequality $ 0<P(A^c)$ holds.
Suppose $(\Omega, \Sigma)$ is a measurable space such that $\Omega$ is a topological space and $\Sigma$ is a Borel $\sigma$-algebra on $\Omega$.
For any such measurable spaces $(\Omega, \Sigma)$, does ...
- 77
3
votes
1
answer
97
views
Does there always exist a probability measure $P$ such that for any $x,y\in X$ there exist neighborhoods $x\in U,y\in V$ for which $0<P(U),P(V)$?
Consider a measurable space $(X, \Sigma)$ such that $X$ is a Hausdorff space and $\Sigma$ is a Borel $\sigma$-algebra on $X$.
For any such measurable space $(X, \Sigma)$, does there exist a ...
- 77
3
votes
1
answer
112
views
When is measurability of a function $f$ equivalent to $f$ being almost everywhere the limit of measurable functions?
I have a suspicion that the following is true: $\def\AAA {\mathcal{A}} \def\BBB {\mathcal{B}} \def\NN {\mathbb{N}}$
Theorem: let $(X,\AAA,\mu)$ be a complete measure space, and $(Y,\BBB)$ a measurable ...
- 5,166
2
votes
0
answers
61
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Prove that infinite union of $\mathcal{F_{i}}$ is not always $\sigma$-algebra
$\mathcal{F_{1}}$ is some sub-algebra and $\mathcal{F_{n+1}}$ is
class of all sets that can be represented as a countable union or
intersection of sets $\mathcal{F_{n}}$
Prove that $\bigcup_{n \in \...
- 21
3
votes
1
answer
46
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Example of an infinite compact measurable space
Let $X$ be a nonempty set with a $\sigma$-algebra $\mathcal{A}$. The notion of $\sigma$-algebra strictly lies between Boolean algebras and complete Boolean algebras. Clearly, $\mathcal{A}$ is a ...
- 1,068