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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Distortion lemma for composition of (distinct) functions expanding on average

I am trying to describe the following dynamics: Let $(T_{\rho})_{\rho \in [0, 1]}$, $T_{\rho}: [-1, 1] \rightarrow [-1, 1]$ be a family map which satisfies: $\forall \, \rho \in [0, 1], \, \exists \,...
Gabriel B. H. Lisboa's user avatar
6 votes
1 answer
330 views

Signed measures and poset inequalities

Consider a triangulated ball $D$, and assume that $\omega$ is an assignment of real weights to the simplices of $D$, including the empty one, such that for every maximal simplex $F$ and every simplex $...
Karim Adiprasito's user avatar
4 votes
2 answers
370 views

A possible measure-theoretic pathology

Let $S$ be a nonempty closed subset of the open unit square $(0,1)^2 = X \times Y$ that has the following "shadow property": For any aligned open square $C = A \times B$ that intersects $S$, ...
pmw's user avatar
  • 41
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0 answers
95 views

Prove that $\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$

I'm folowing the proof of corollary 1.8 page 5 of Mörters - Sample path properties of Brownian motion. I want to show that $$\forall x,y \in \mathbb{R}^d , P_x\{y\in B\mathopen]0,1]\}=0$$ where $B$ is ...
sara's user avatar
  • 11
1 vote
0 answers
63 views

Benjamini-Schramm convergence: convergence on metric balls implies weak convergence?

In the famous paper by Benjamini-Schramm 2001, they consider the space of rooted graphs with uniformly bounded degrees, this space modulo rooted graph isomorphism is denoted by $\mathcal X$. This ...
Fredy's user avatar
  • 492
-1 votes
1 answer
298 views

The category Prob of finite measure spaces does not admit all products [closed]

I am currently working in a category called Prob which has objects which are finite measure spaces and morphisms which are measure preserving maps between the spaces. A map $f:X\to Y$ is measure ...
Maat's user avatar
  • 91
5 votes
1 answer
381 views

Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?

Let $\varphi: \mathbb{R}^d \to \mathbb{R}$ be a convex function. The subdifferential of $f$ at $x$ is defined as $$ \partial \varphi (x) := \{z \in \mathbb{R}^d : \varphi(y) \geq \varphi(x) + \langle ...
Akira's user avatar
  • 851
19 votes
1 answer
3k views

Is the area of the Mandelbrot set known? [duplicate]

The Mandelbrot set has an area; is it known exactly? If so, how, and what is the value? If not, why is this a hard question to answer?
user6873235's user avatar
15 votes
4 answers
1k views

Steinhaus theorem and Hausdorff dimension

Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...
Piotr Hajlasz's user avatar
1 vote
0 answers
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Symmetry of the isoperimetric profile

Given a probability measure $\mu$ on a metric space $(X, \mathsf{d})$, the $(\mu-)$Minkowski content of a set $A$ is defined as $$\mu^+ (A) := {\lim\inf}_{r \to 0^+} \frac{\mu ( A_r \setminus A)}{r},$$...
πr8's user avatar
  • 688
19 votes
2 answers
1k views

Forward definition of measurability

Is there a characterisation of measurability in a forward way similar to the closure characterisation of continuity? A function $f\colon X \to Y$ is continuous if equivalently: If $G\subset Y$ is ...
Moritz Schauer's user avatar
0 votes
1 answer
213 views

Is there a lop-sided permutation $\pi:\mathbb{N}\to\mathbb{N}$? [closed]

For any $A\subseteq \mathbb{N}$ we let the (lower) density of $A$ be defined by $$d(A) = \liminf_{n\to\infty}\frac{|A\cap\{0,\ldots,n\}|}{n+1}.$$ If $\pi:\mathbb{N}\to\mathbb{N}$ is a permutation (...
Dominic van der Zypen's user avatar
2 votes
1 answer
206 views

Invariant measure of geodesic flow on unit tangent bundle of a modular surface

This is a paper written by Series "THE MODULAR SURFACE AND CONTINUED FRACTIONS". I want to know about above construction natural invariant measure $\mu$ for the geodesic flow on $T_{1}M$ ...
user473085's user avatar
1 vote
0 answers
47 views

Is $L^p_\text{loc} (Y)$ dense in $(L^0(Y), \hat \rho)$?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(Y, \...
Analyst's user avatar
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9 votes
1 answer
314 views

A topological characterisation of a.e. continuity

We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
Nate River's user avatar
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1 vote
0 answers
50 views

Is there $r>0$ such that the norm $[f]_r:=\sup_{n \ge 1} \|1_{B(y_n, r)} f\|_{L^p}$ is equivalent to $\|f \|:=\sup_{y \in Y}\|1_{B(y,1)}f\|_{L^p}$?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(E, |\...
Analyst's user avatar
  • 647
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0 answers
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Is the embedding $i: (L^p_\text{loc} (Y), \| \cdot \|_{L^p_\text{loc}}) \to (L^0(Y), \hat \rho)$ continuous or Borel measurable?

Below we use Bochner measurability and Bochner integral. Let $(Y, d)$ be a separable metric space, $\mathcal B$ Borel $\sigma$-algebra of $Y$, $\nu$ a $\sigma$-finite Borel measure on $Y$, $(Y, \...
Analyst's user avatar
  • 647
2 votes
2 answers
73 views

Reference for Wiener type measure on $C(T)$ when $T$ is open

I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
Kiyoon Eum's user avatar
0 votes
0 answers
109 views

The set of measurable functions together with convergence in measure is a completely metrizable abelian topological group

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple ...
Analyst's user avatar
  • 647
1 vote
0 answers
29 views

Can we modify this extended pseudometric such that its convergence is equivalent to that in measure?

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete $\sigma$-finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple ...
Analyst's user avatar
  • 647
2 votes
1 answer
187 views

Relationship between Baire sigma algebra and Borel sigma algebra of an uncountable product

I've been trying to understand various questions to do with sigma algebras on uncountable product spaces. Let $T$ be an uncountable set and for each $t \in T$, let $\Omega_t$ be a topological space. ...
SBK's user avatar
  • 1,141
3 votes
0 answers
137 views

Which is the smallest $\sigma$-algebra that contains all analytic sets?

Let $X$ be a polish space. Is the smallest $\sigma$-Algebra containing all analytic sets of $X$ (i.e. all subsets $A \subset X$ which are the continuous image of a polish space) the $\sigma$-algebra ...
Joris Wk's user avatar
  • 233
1 vote
0 answers
63 views

Is the projection of an universally measurable set again universally measurable?

Let $(X,\mathcal{A})$ be a measurable space and $(Y,\mathcal{B}(Y))$ be a polish space together with the Borel-$\sigma$-Algebra. There is a Theorem that states: The projection $\pi_X(B)$ of every ...
Joris Wk's user avatar
  • 233
0 votes
0 answers
85 views

Assume $f(x, \cdot) \in L^p_{\text{loc}} (Y)$ for a.e. $x \in X$. Is $F: X \to L^p_{\text{loc}} (Y), x \mapsto f(x, \cdot)$ Bochner measurable?

Below we use Bochner measurability and Bochner integral. Let $T>0$ and $p \in [1, \infty)$. Let $X :=[0, T]$ and $Y:= \mathbb R^d$. Let $L^p_{\text{loc}} (Y)$ be the space of measurable functions $...
Akira's user avatar
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1 vote
1 answer
174 views

Is there a version of dominated convergence theorem for local $L^p$ spaces?

Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
Akira's user avatar
  • 851
3 votes
2 answers
142 views

Mollifying a measure without changing its marginals

Is there a reasonable/canonical way to mollify a Borel probability measure without changing its marginals. Let $\pi \in \mathcal{P}(\mathbb{R}^2)$ with marginals $\mu,\nu$. I want to smooth out $\pi$ ...
almosteverywhere's user avatar
1 vote
0 answers
78 views

Is this metric on the space of $\mu$-measurable functions complete?

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ be a complete finite measure space, $(E, | \cdot |)$ a Banach space, $S (X)$ the space of $\mu$-simple functions ...
Analyst's user avatar
  • 647
0 votes
2 answers
84 views

How to construct this sequence that converges a.e. in product measure and that has a very particular form?

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
Akira's user avatar
  • 851
3 votes
3 answers
474 views

Solving interval problems without outer measure

Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ? Problem 1 If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
Ross Ure Anderson's user avatar
0 votes
1 answer
144 views

For $\mu$-a.e. $x \in X$, the sequence $(f_n(x, \cdot))_n$ is Cauchy in $L^1 (Y)$. Then $(f_n)$ is Cauchy in $L^1 (X \times Y)$

Below we use Bochner measurability and Bochner integral. Let $(X, \mathcal A, \mu)$ and $(Y, \mathcal B, \nu)$ be complete $\sigma$-finite measure spaces, $(E, | \cdot |)$ a Banach space, $S (X)$ the ...
Akira's user avatar
  • 851
0 votes
2 answers
120 views

Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?

Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
Akira's user avatar
  • 851
1 vote
1 answer
140 views

Resources to understand Lebesgue measure of Brownian motion's path [closed]

[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47] Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
sara's user avatar
  • 11
3 votes
2 answers
400 views

A general inequality for KL divergence of functions of variables

The question concerns a very general setting and a very general inequality about KL divergence. I'm writing this thread to verify whether my intuition is correct. Let $E_1, E_2$ be two measurable ...
Daniel Goc's user avatar
1 vote
1 answer
229 views

What is the measure of two sets which partition the reals into subsets of positive measure?

This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$. (In ...
Arbuja's user avatar
  • 1
4 votes
1 answer
139 views

Is the set of clopen subsets Borel in the Effros Borel space?

Let $X$ be a Polish space and $\mathcal{F}(X)$ the set of closed subsets of $X$ endowed with the Effros Borel structure, generated by sets of the form $\{F\in \mathcal{F}(X):F\cap U\neq \emptyset\}$, ...
Iian Smythe's user avatar
  • 3,001
1 vote
0 answers
79 views

Borel structure/sets coming from strong operator topology vs norm topology

Let $X, Y$ be Banach spaces. Moreover, let $\mathcal{L}(X,Y)$ be the space of bounded linear operators equipped with the standard operator norm topology, and $\mathcal{L}_{\mathrm{s}}(X,Y)$ the same ...
Marek Kryspin's user avatar
1 vote
1 answer
81 views

Approximating a family of measurable functions

Let $X$ be a set of $N>0$ elements (with the counting measure) and consider a family of measurable functions $f_i:X\to [0,1]$, for $i\in \mathbb N$. Any function $f_i$ can be seen as a point in the ...
manifold's user avatar
  • 299
0 votes
0 answers
30 views

Concerning the conversion of an essential supremum to a pointwise estimate

In the following paper : Chen, Zhen-Qing; Kumagai, Takashi; Wang, Jian, Stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms, J. Eur. Math. Soc. (JEMS) 22, No. 11, 3747-...
Sarvesh Ravichandran Iyer's user avatar
0 votes
1 answer
142 views

Billingsley convergence of probability measures - inequality used in Theorem 2

On Page 8, Billingsley defines $f(x)=(1-\rho(x,F)/\epsilon)^{+}$ where $\rho(x,F)$ is the metric distance from the set $F$. He then states $|f(x)-f(y)|\leq \rho(x,y)/\epsilon$ and goes on to use this ...
Simon's user avatar
  • 9
1 vote
0 answers
138 views

Polish spaces and analytic sets

Can we conclude that an analytic subset $A$ of a Polish space $X$ is also Polish? Let $\mathcal{M}(R^d)$ denotes the family of Borel probability measures on $R^d$ equipped with the Lévy-Prokhorov ...
B-S's user avatar
  • 39
1 vote
1 answer
194 views

What is convergence in distribution of random variables taking values in a non-metrizable product space?

Let $F = (F(x) : x \in \mathbf{R}^n)$ be a family of $\mathbf{R}^k$-valued random variables indexed by $\mathbf{R}^n$ (to be clear there is a single probability space $(\Omega,\Sigma,\mathbf{P})$ such ...
SBK's user avatar
  • 1,141
5 votes
1 answer
358 views

Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane?

Suppose $\lambda^{*}$ is the Lebesgue outer measure. Question: Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where: The range of $f$ is $\mathbb{R}$ For all real $x_1,x_2,y_1,y_2$, where $...
Arbuja's user avatar
  • 1
2 votes
1 answer
151 views

Finding an explicit & bijective function that satisfies the following properties?

Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$. Question: Does there exist an explicit ...
Arbuja's user avatar
  • 1
2 votes
0 answers
121 views

Measure algebra for a family of probability measures

Let $(X,B,P)$ be a probability space, $I_P$ the $\sigma$-ideal of $P$-null sets and \begin{align} B_P = B \ltimes I_P &= \{ A \mathbin{\triangle} N \mid A \in B, N \in I_P \} \end{align} the ...
Packo's user avatar
  • 188
0 votes
1 answer
124 views

An integral Minkowski inequality for the quasi-Banach case?

The so called integral Minkowski inequality claims that for suitable positive functions $f$ defined on the product of two measure spaces $ (X\times Y, \mu \times \nu) $ and $p\geq 1$ we have that $$ \...
an_ordinary_mathematician's user avatar
4 votes
0 answers
162 views

Finding balls with big measure

Let $(X,d)$ be a compact metric space $n \in \mathbb{N}$ and $\mu$ a finite Borel measure. Suppose there exists $\delta, R>0$ such that for all $0<r<R$. $$\mu(B(x,r)) < \delta r^n.$$ Under ...
Denis Marti's user avatar
2 votes
0 answers
46 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
5 votes
1 answer
163 views

Lifting a semicommutative von Neumann algebra into an algebra of operator-valued functions

Given a complete probability space $\Omega$ and a von Neumann algebra $\mathcal M$, it is well-known that the tensor product von Neumann algebra $\mathrm L^\infty(\Omega)\overline\otimes\mathcal M$, ...
P. P. Tuong's user avatar
3 votes
2 answers
252 views

Representing measurable map to compact space as a continuous map

Let $\Omega$ be a measurable space equipped with a $\sigma$-ideal $\mathcal{N}$ (though of as the "null sets"). Define the compact Hausdorff space $$ \tilde{\Omega} := \mathrm{Spec}(L^\infty(...
user avatar
9 votes
3 answers
789 views

How to prove that the Lebesgue $\sigma$-Algebra is not countably generated?

How to prove that there can't exist a countable set $\{A_1,A_2,\dots\}\subset \mathcal{L}(\mathbb{R})$ (where $\mathcal{L}(\mathbb{R})$ denotes the family of all Lebesgue measurable sets) such that $\...
Joris Wk's user avatar
  • 233

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