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# 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 ...
1answer
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 ...
1answer
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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1answer
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### 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 ...
1answer
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 ...
0answers
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 ...
1answer
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### 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)) \...

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