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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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### Deduce that a function is zero on interval $[0,M]$

I have been thinking about this for the last few days but I was not able to produce a definitive answer. Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
1 vote
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### Strong subobject classifier for measurable spaces

Does the category of measurable spaces have a strong subobject classifier (specifically $2 = (\{0,1\}, \{\varnothing, \{0,1\}\})$? I would think the situation could be analogous to $\mathsf{Top}$, ...
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### Følner sequences of the integers

Definition: Let $G$ be a group. For $g\in G$ and a subset $F\subseteq G$ fix the notation $gF:=\{gf\mid f\in G\}$. A sequence $(F_{i})_{i\in\mathbb{N}}\subseteq G$ is called a Følner sequence if \...
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### Does the existence of regular conditional measure follow from that of regular conditional distribution?

For more succinct description, I use the following abbreviations, i.e., RCPD: Regular conditional probability distribution. RCPM: Regular conditional probability measure. First are definitions 10.4....
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### Push-forward of uniform measure and uniqueness

On a standard Borel (or Polish) space $X$, any probability measure $\mu$ is the push-forward of the uniform measure on $[0, 1]$ under some $f : [0, 1] \to X$. This $f$ is not unique in general. ...
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### Kolmogorov $\epsilon$-entropy, $n$-width, and $\epsilon$-capacity and applications

What is the relationship between Kolmogorov $\epsilon$-entropy, Kolmogorov $n$-width, and Kolmogorov $\epsilon$-capacity of a set $M$ in a metric space $X$? (The $\epsilon$-capacity here is the ...
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### Are nonatomic probability measures on a Banach space nicely shrinking a.e?

Let $\mu$ be a nonatomic probability measure on a Banach space $X$. Is it true that for $\mu$ a.e. $x \in X$, the function $g_x: (0, \infty) \to \mathbb R$ given by $$g_x (r) := \mu(B_r (x))$$ is ...
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### Disintegration of measures: a confusion about an existence proof from a lecture note

I'm reading a proof of Theorem 2.25 below from this note. First, we recall a definition and a theorem, i.e., Theorem 2.25 (Disintegration). Let $\left(Z, d_Z\right)$ and $\left(X, d_X\right)$ be ...
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### Deducing differential equations from a time-continuous Markov chain via its rate matrix

I have only basic level knowledge of probability theory and I am researching in a different field. So please do not be too harsh on me if my question turns out to be silly. Let $(X, \Sigma, \mu)$ be a ...
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### Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?

I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
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### Weak convergence of random measures generated by non-negative martingales?

If I have a sequence of non-negative continuous martingales $(M_n(x))_{n\ge 1}$ on $x\in[0,1]$, i.e. for each fixed $n$, $M_n:[0,1]\to[0,\infty)$ is a continuous process, and for each fixed $x\in[0,1]$...
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### How to find the point at minimal average distance of a given measure

Given a compactly supported probability measure $m$ on $\mathbb{R}^n$, we can define its average distance to a point $x$ as $\int_\mathbb{R^n}d(x,y)dm(y)$. In this question I found that for a given ...
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### Equivalence between Lebesgue integrable and Riemann integrable functions

As the title says, for every Lebesgue integrable function $f:\mathbb{R}\to\mathbb{R}$ is there a Riemann integrable function $g:\mathbb{R}\to\mathbb{R}$ such that $f=g$ almost everywhere? For example, ...
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### Continuity of the traffic intensity

Note: We say that a Borel measure on $\mathbb R^n$ has a continuous density if it is absolutely continuous with respect to Lebesgue measure and has a continuous Radon Nikodym derivative. Let $\mu$ be ...
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### Transforming two smooth densities to the same density

I am looking for an example of the following: Find a bijective, differentiable function $f$ and continuous probability density functions $q_1\ne q_2$ such that $f_*q_1=p=f_*q_2$, where $f_*$ is the ...
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### A measure on the group of homeomorphisms of $\mathbb T^2$

Let us consider the group of measure-preserving homeomorphisms of $\mathbb T^2$ (with transformations identified if they agree almost everywhere) called $G[\mathbb T^2, \mathcal L^2]$. We shall ...
1 vote
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### Building random homeomorphisms of the torus $\mathbb T^2$

In https://arxiv.org/abs/0912.3423, a family of random homeomorphisms of the circle is constructed. Main Question: Can the construction be generalized to higher space dimensions, e.g. to $\mathbb T^2$?...
1 vote
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### Building random homeomorphisms of the circle

Given a positive Borel measure without atoms $\tau$ on the circle $\mathbb T =\mathbb R /\mathbb Z =[0,1)$ , in https://arxiv.org/abs/0912.3423 a homeomorphism $h:[0,1)\to [0,1)$ is defined as \...
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### If every point is a Lebesgue point of $f$, is $f$ continuous a.e.?

Let $f: \mathbb R^n \to \mathbb R$ be a locally integrable function. Question: Suppose every point $x \in \mathbb R^n$ is a Lebesgue point of $f$. Does it follow that $f$ is continuous almost ...
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### Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of Baire for every $F$?

Let $F\subset \Bbb{R}$ intersect every closed uncountable subsets of $\Bbb{R}$. Does there exist $f:\Bbb{R}\to \Bbb{R}$ additive onto function such that $f(F) \subset \Bbb{R}$ has the property of ...
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### Kolmogorov 0-1 law counter examples for almost independent variables

According to Kolmogorov 0-1 law, if $\left(X_{i}\right)_{i=1}^{\infty}$ are independent, then the tail sigma algebra is trivial. I want to construct such variables which are "almost independent&...
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### Why impossible events have some drawbacks or pathologies in probability theory?

It is said by Halmos, P.R.; in "Lectures on ergodic theory" "Many of the difficulties of measure theory and all the pathology of the subject arise from the existence of sets of measure ...
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### Does $\sum_{|S| \text{ even}} f(S) \le \sum_{|T| \text{ even}} f(T)$ hold for all nondecreasing submodular functions f?

Let $f : 2^n \to \mathbb{R}$ be a nondecreasing submodular function, where $2^n$ is the powers of $\{1, \dots, n\}$. Here nondecreasing means that $f(S) \le f(T)$ for all $S \subseteq T$. And ...
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### Malliavin-Shavgulidze (type) measures on the group of measure-preserving invertible maps on $\mathbb T$?
The Malliavin-Shavgulidze measures on $\operatorname{Diff}^{1}(I)$ (with $I$ an interval of $\mathbb R$) are defined as the image $W_{\sigma} \circ f^{-1}$ of the Wiener measure $W_{\sigma}$ with ...
From Halmos, Paul R., In general a measure preserving transformation is mixing, Ann. Math. (2) 45, 786-792 (1944). ZBL0063.01889. the following result is known: Let $(E,\Sigma, \mu)$ be a measure ...