# Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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### Canonical measure on Pareto front?

Suppose $F:\mathbb R^d\to\mathbb R^D$ encodes a set of objective functions on $\mathbb R^d$, and take $\mathcal S\subseteq \mathbb R^D$ to be the Pareto front of $F$. Under some genericity/smoothness ...

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68 views

### Measure of real numbers with converging average over binary digits

Consider the unit interval $[0,1]$, and by digits of $x\in[0,1]$ I mean its binary digits after the separator with no 1-period.
If $x_1,x_2,x_3,...$ are the digits of $x$, then consider the $k$-th ...

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182 views

### How can the same polytope have three different volumes?

I'm quite new to geometry and I came across the idea that the same convex polytope can have at least three different volumes.
Consider the permutohedron, formed by the convex hull of the n! points ...

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21 views

### Predictable Projection of a Stopped Process (Typo in Jacod & Shiryaev?)

Given a filtered probability space $( \Omega, \mathcal{F}, (\mathcal{F}_t)_t, \mathbb{P})$ and an $\mathcal{F} \otimes \mathcal{B}(\mathbb{R}_+)$-measurable bounded process $X: \Omega \times \mathbb{R}...

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141 views

### Must a locally compact, second countable, Hausdorff space support a Radon measure?

Let $X$ be a locally compact, second countable and Hausdorff space, must there be a Radon measure on $X$ whose support is $X$?
The motivation for this question comes from Anton Deitmar's paper On ...

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59 views

### Volume form induces Borel measure: proof verification [migrated]

Disclaimer: I asked this question already on the regular mathematics site here, but to no avail, even with a bounty. I think answering said question is still of value.
Proposition. Let $M$ be a ...

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73 views

### Probability space with countable subset such that every subset of positive measure meets the subset

Let $(X, \mathcal F, P)$ be a probability space.
Question
What kind of condition is this: there exists a sequence $(a_n)_n \subseteq X$ such that
$\forall$ measurable $A \subseteq X$, $P(A) >...

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136 views

### Is every closed subset of finite measure contained in an open subset of finite measure?

Could someone will verify my statement: For every locally finite Borel measure on metric space and closed set $F$ with finite measure, there exists open set $U$ such that $F \subset U$ and $U$ has ...

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47 views

### Existence of sequence of measurable sets with prescribed densities

Consider Lebesgue measure $m$ on $[0, 1]$. Fix a countable sequence $a_i, 0 < a_i < 1$ such that $\sum_i a_i = 1$. Is there a sequence of disjoint measurable subsets of $[0, 1]$, $E_i$ whose ...

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32 views

### Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?

Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$.
I'm looking to construct some kind of dyadic cube decomposition or ...

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91 views

### Is there a density theorem for Banach measure?

Fix a finitely additive measure $\mu$ on $\mathbb R^2$ that is consistent with the Lebesgue measure. Does Lebesgue's density theorem hold for such a $\mu$, i.e., is it true that for every $A$ we have $...

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57 views

### Regularity of locally finite Borel measure

Do you know any proof that locally finite Borel measure on metric space is regular ? I found many proofs only for finite Borel measure, but it's not satisfies me. Or maybe do you know any books or ...

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196 views

### Attractors in random dynamics

Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...

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46 views

### Union of 2-dilation of a Lebesgue measurable set is real line R?

Def : Frame Wavelet Set
Let $E$ be a Lebesgue measurable set of finite measure. Define $\psi \in L^{2}(R)$ by $\hat{\psi}=\frac{1}{\sqrt{2\pi}} \chi_{E}$, where $\hat{\psi}$ is Fourier Transform of $\...

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31 views

### What is the Wiener measure of the set of curves with given Hölder constant on a Riemannian manifold?

Let $M$ be a connected Riemannian manifold and $x_0 \in M$. For $0 < \alpha < \frac 1 2$, let
$$H = \{ c : [0,1] \to M \mid c(0) = x_0 \text{ and } \exists C>0 \text { s.t. } d(c(s), c(t)) \...