All Questions
Tagged with borel-sets lebesgue-measure
9 questions
0
votes
0
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112
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Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
1
vote
1
answer
692
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Is the point-wise limit of simple functions a measurable function?
Let $X$ and $Y$ be topological spaces. By a simple function $\phi: X\to Y$ we mean a finite range Borel measurable function.
Q. Is the point-wise limit of a sequence of simple functions a Borel ...
- 4,058
15
votes
2
answers
530
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Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets
Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets.
Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
- 807
5
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0
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363
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Computing the infinite dimensional Lebesgue measure of "cubes"
There is no Lebesgue measure in infinite dimensions—this slogan is familiar to every student interested in analysis. One possible, precise statement of this result may be as follows: if $X$ is an ...
- 9,330
14
votes
1
answer
735
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Does there exist a non-zero signed finite borel measure which is zero on all balls?
Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, signed measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (...
- 345
16
votes
2
answers
1k
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Is it known how the Sigma Algebra generated by Jordan measurable sets compares to universally measurable sets and analytic sets?
Unlike the collection $L$ of Lebesgue measurable sets, the collection $J$ of Jordan measurable sets do not form a Sigma algebra. (A set is Jordan measurable if and only if its characteristic function ...
- 4,587
3
votes
1
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568
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Does there exist a Lebesgue nonmeasurable set $E$ in $\mathbb{R}$ satisfies that $E\cap A$ is a Borel null set for every Borel null set $A$?
Let $\mathcal{B}_{\mathbb{R}}$ be the Borel $\sigma$-algebra on $\mathbb{R}$ and $\mu_L$ be the Lebesgue measure on $\mathbb{R}$.
Define a new $\sigma$-algebra $\mathcal{B}_0$ as follows:
$$\mathcal{...
- 1,997
2
votes
2
answers
233
views
Is the domain of symmetric derivative borel set?
Let $\mu$ be the $n$-dimensional Lebesgue measure and $\lambda$ be a complex Borel measure on $\mathbb{R}^n$.
Let $S$ be the set of points $x\in \mathbb{R}^n$ where $\lim_{r\to 0} \frac{\lambda (B(x,...
- 337
3
votes
2
answers
236
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Can a continuous surjection from a Hilbert cube to a segment behave bad wrt Lebesgue measures?
Suppose $\hat{I}=[0,1]^\mathbb{N}$ is a Hilbert cube and $I=[0,1]$.
Consider Lebesgue measures $m_1$ and $m_2$ on $\hat{I}$ and $I$ correspondingly. By Lebesgue measure on the Hilbert cube I mean the ...
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