# Questions tagged [measure-theory]

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

2,701
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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 $[0,M]$ (...

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1
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102
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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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### On a certain deterministic integral related to Tanaka’s formula

Note: We define the signum function, $\text{sgn}$ by $\text{sgn}(x) = 1$ if $x \geq 0$, and $-1$ otherwise.
Suppose $f: [0, \infty) \to \mathbb R$ is continuous and of locally bounded variation, with $...

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### Arzelà–Ascoli for equi-Lebesgue continuous functions

Given a measurable subset $A$ of $[0, 1]$, a sequence of functions $f_n: [0, 1] \to \mathbb R$ is said to be equi-Lebesgue continuous on $A$ if for every $x \in A$, and $\varepsilon > 0$, there ...

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### Formulation of $p$-adic Haar measure decomposition

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\vol{vol}\DeclareMathOperator\diag{diag}$Suppose:
$F$ is a non-archimedean local field,
$\mathcal{O} \subset F$ its ring of integers,
$\pi \in \mathcal{...

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138
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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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### Uniform bound on the measure of $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ if $\Omega$ is an open bounded set with Lipschitz boundary

Let $\Omega \subset \mathbb R^d$ be an open bounded set with Lipschitz boundary. Let us consider $\Omega_\delta = \Omega \cap \delta\mathbb Z^d$ for $\delta >0$. I want to say that the measure of $\...

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### Continuity of Wiener measure on open balls

Let $\mu$ be the Wiener measure on $C_0 [0, T]$, the space of continuous functions starting at $0$, under the sup norm.
Question: Is it true that the function $r \mapsto \mu(B_r(x))$ is continuous in $...

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184
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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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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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### Restriction to dense subset of functions whose graph is dense

Let $f: [0, 1] \to \mathbb R$ be a measurable function. A function $g: [0, 1] \to \mathbb R$ is said to be a condensation limit of $f$ if $g$ is continuous and agrees with $f$ on a dense subset of $[0,...

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### "N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$

Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...

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### Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...

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### Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$

Now we fix an ultrafilter of $\mathbb{N}$ that contains the cofinite filter, consider a hyperreal field ${}^{*}\mathbb{R}$. Let $\varepsilon$ be a positive infinitesimal. We doubt that a power series ...

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### Is the domain space in Lusin's theorem required to be Hausdorff?

I'm reading a general version of Lusin's theorem, i.e.,
If $\mu$ is a finite Radon measure on $X$, and $Y$ is a second countable topological spaces, then for any Borel-measurable function $f:X\to Y$ ...

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### For a vector field $f$ and a measure $\mu$, does conservativity $\mu$-almost everywhere have a sense?

$\DeclareMathOperator\Jac{Jac}$
Let $\Omega$ an open star shaped subset of $\mathbb{R}^d$ and $f : \Omega \rightarrow \mathbb{R}^d$ a differentiable vector field. For $x \in \mathbb{R}^d$, we denote ...

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### Can the Banach-Tarski paradox or Tarski's circle-squaring problem be done with hinges?

It is known for both the Banach-Tarski paradox and Tarski's circle-squaring problem that you can finitely partition the starting configuration, then continuously move these pieces (without ...

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### What is the limit of a helix as the frequency tends to infinity?

Consider the helix parametrized by $r(t) = (\cos(\omega t), \sin(\omega t), t)$, for a given $\omega > 0$, and $t \in \mathbb{R}$. How can we interpret the limit as $\omega \to \infty?$
My initial ...

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### Is (the generalised) Sard's theorem optimal?

As mentioned in this question (https://math.stackexchange.com/questions/416607/show-that-fc-has-hausdorff-dimension-at-most-zero/446049#446049), in 1965 Sard proved the following result (paraphrased ...

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### $\Psi$ in finite Wiener–Itô Chaos implies existence of continuous representative on neighborhood of Cameron–Martin space?

Let $(\Theta, H, \mu)$ be an abstract Wiener space, i.e. let $(\Theta, \lVert \cdot \rVert_{\Theta})$ be a separable Banach space, let $(H, \langle \cdot, \cdot \rangle_{H})$ be a separable Hilbert ...

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### Equivalent definition of the Kantorovich-Fisher-Rao distance

I am reading this paper
"A JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows"
(https://arxiv.org/abs/1602.04457)
and in the proof of Proposition 2.2, basically, if the measure ...

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### If the measure theoretic boundary is closed must it coincide with the topological boundary?

$\DeclareMathOperator\Int{Int}\DeclareMathOperator\Ext{Ext}$Suppose $E\subset\mathbb{R}^n$ is a set of finite perimeter and suppose that the measure theoretic boundary $\partial^*E=\mathbb{R}^n\...

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### Does approximately Fréchet differentiable imply approximately Gateaux differentiable?

In what follows, $\mu^n$ denotes $n$-dimensional Lebesgue measure, and $B_r(x)$ is the open ball with radius $r$ centered at $x$.
In elementary calculus, if we have a function $f : \mathbb{R}^n \...

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### Ergodic diffeomorphisms of the circle

From the paper
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 ...

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1
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180
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### Large cardinals and measurability in $L(A)$

Under suitable large-cardinal assumptions, in the inner model $L(\mathbb R)$ one can have $\omega_1$ and $\omega_2$ measurable (this follows from determinacy).
I was wondering if it is possible to ...

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### Convex ordering of measures that are obtained by different push-forwards of a same measure

Suppose that we have a probability measure $\rho$ which is supported on $\mathbb{R}^d$ and absolutely continuous w.r.t. the Lebesgue measure. Take two vector fields $F, G : \mathbb{R}^d \rightarrow \...

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1
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### Cameron-Martin space of product space

Suppose you have Banach spaces $\mathcal B_\alpha$ where $\alpha$ is in some index set $I$. Let $\mu_\alpha$ be Gaussian measures on $\mathcal B_\alpha$ with Cameron-Martin spaces $\mathcal H_{\mu_\...

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### The Borel sigma-algebra of a product of two topological spaces

The following problem arose while trying to justify some "known results" in abstract harmonic analysis on noncommutative groups, for which I couldn't find explicit statements in the ...

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### Existence of a limit of alpha-difference quotient for Hölder functions

Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...

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### Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions

We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...

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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 ...

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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 $\mathbb T^2$?...

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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 ...

2
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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 ...

2
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### Weakly mixing diffeomorphism

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 ...