# Tagged Questions

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

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### question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see: http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
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### Transitive closure of balanced bounded mass transport

Given two $\sigma$-finite measures $\mu$ and $\nu$ on $\mathbb{R}^n$, write $\mu \sim \nu$ iff there exist countable decompositions $\mu = \mu_1 + \mu_2 + \cdots$ and $\nu = \nu_1 + \nu_2 + \cdots$ ...
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### Is a sigma-finite Borel measure over $\mathbb R$ determined by its values on the continuous functions? [closed]

Suppose that $\mu$ and $\nu$ are sigma-finite measures on the Borel sigma-algebra over $\mathbb R$ such that $\int_{\mathbb R}f\,d\mu=\int_{\mathbb R}f\,d\nu$ for all nonnegative continuous functions ...
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### Is every set of small measure contained in an open set of small measure with null boundary?

Let $\lambda( \cdot )$ denote Lebesgue measure on $[0,1]$. Let $(A_n)_{n=1}^\infty$ be a decreasing sequence of Borel subsets of $[0,1]$ such that $\bigcap_{n=1}^\infty A_n = \emptyset$. Given ...
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### Hausdorff measure of the graph

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure? Of course if such ...
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### Criteria for Compactness of a Closed in $L^2$ Spaces [closed]

$(X, \mathcal{B}, \mu)$ is a measure space. Is there any well-known criteria for compactness of a closed set in $L^2(X, \mu)$? If the answer is negative what about $L^2(\mathbb{R}^n,\mu)$(in this ...
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### How can dimension depend on the point?

Let $M$ be a metric space. For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension. For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
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### Fubini's theorem without completeness or $\sigma$-finiteness conditions

The usual Fubini's theorem(see the Wikipedia article for example) assumes completeness or $\sigma$-finiteness on measures. However, I think I came up with a proof of the Fubini's theorem without those ...
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### Reading Ratner's paper “Ragunathan's conjectures for SL(2,R)”

Hello everyone (interested), I am trying to read Marina Ratner's paper "Ragunathan's conjectures for $SL_{2}(R)$" (Israel Journal of Mathematics 80 (1992), 1-31). There is a claim right at the end of ...
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### Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be ...
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### Injective inclusion map from RKHS function space to $L_p(\mu)$

Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$. At a certain part in a proof I ...
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### Baire Category Theorem Application

In Antoine Henrot Michel Pierre - Variation et optimisation de formes, Une analyse geometrique, a book I'm studying I found an interesting problem. The problem is listed below. The first 3 points of ...
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### Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...
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### Function and its Gradient with Prescribed Norms

I'm not sure if the following question is too elementary for Mathoverflow. I'm sorry if it is the case. Question: Let $n\in\mathbb{N}$ and let $1\leqslant p<\infty$. Let $\alpha,\beta>0$. ...
I am following these lecture notes: Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$. Corollary ...