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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Comparing two $\sigma$-algebras

Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$. Q. For which ...
ABB's user avatar
  • 3,898
4 votes
2 answers
432 views

Lifting back the induced invariant measure / general version of Kac's formula for occupation times

Let $T$ be a conservative measure preserving (non-invertible!) transformation of a measure space $(X, \mathscr{F}, m)$ with infinite measure $m$. Let $A \in \mathscr{F}$ be such that $X = \cup_{k=0}^\...
Vlad Vysotsky's user avatar
6 votes
1 answer
643 views

On the failure of extending a probability measure on uncountable $\Omega$

It is a well known fact that if $(\Omega, \mathcal{F}, P)$ is a probability triple and $\{A_i : i < k\}$ is a finite collection subsets of $\Omega$, then there is a $P' \supset P$ and $\mathcal{F'} ...
Zoorado's user avatar
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8 votes
1 answer
230 views

Is there a non-atomic finite positive measure in the plane, of which uncountably many projections have atoms?

I would like to know whether or not there exists a finite probability measure $\mu$ on $\mathbb R^2$ which has no atoms, but such that there exists an uncountable set $A\subset \mathbb S^1$, such that ...
Mircea's user avatar
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2 votes
1 answer
351 views

Daniell integral vs. Borel measure

Let $X$ be a locally compact, Hausdorff, topological space and denote by $\mathcal B_+(X)$ the collection of all Borel-measurable functions from $X$ to $[0,+\infty]$ (extended positive reals). ...
Ruy's user avatar
  • 2,233
6 votes
2 answers
588 views

Interpolation space between $L^1\cap L^2$ and $L^1$

In the paper of Bourgain, the way equation (3.78) is deduced from (3.69) and (3.76) seems via the following interpolation result. Let $(X,\mu)$ and $(Y,\nu)$ be two measure spaces and let $T$ be a ...
shrinklemma's user avatar
1 vote
0 answers
112 views

Constructing a measure from an integral

A well known Theorem by Riesz says that every continuous linear functional on the space $C(X)$ of all complex valued continuous functions on the compact space $X$, is given as the integral against a ...
Ruy's user avatar
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9 votes
1 answer
370 views

Comparing two $\sigma$-algebras on $B(\ell^1)$

Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow $$w-\lim T_i=T \Longleftrightarrow \...
ABB's user avatar
  • 3,898
5 votes
2 answers
401 views

Do a Hausdorff space and its associated completely regular space have the same Borel subsets?

Let $(X,T)$ be a Hausdorff topological space. Let $C_b(X)$ be its algebra of continuous bounded functions. Let $T'$ be the initial topology on $X$ given by $C_b(X)$. It is known that $T=T'$ if and ...
Alex M.'s user avatar
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6 votes
3 answers
422 views

Point-wise limit of finite valued functions

Let $X$ be a second countable topological vector space. Does there exist any sequence of finite valued functions $f_n\colon X\to X$ converging point-wise to the identity mapping on $X$?
ABB's user avatar
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1 vote
1 answer
119 views

Convergence of $L^p$ of approximation

Let $f \in L^p(\mathbb R^n)$ be given. Consider a partition of rectangles $I_{ij}:=[x_i,x_{i+1}]\times [x_j,x_{j+1}]$ of $\mathbb R^2.$ Then, we may define the coefficients $$\alpha_{ij}= \frac{1}{\...
Clement G.'s user avatar
6 votes
2 answers
468 views

Continuity of disintegrations

Suppose that $\pi:Y\to X$ is a continuous surjection from one compact metric space to another. Given a regular probability measure $\mu$ on $Y$ with pushforward measure $\nu:=\pi^*\mu$, it is known ...
Isaac's user avatar
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0 answers
171 views

Regularity of level sets of Sobolev derivatives

I am interested in the regularity of the sets $$U_{\lambda}:=\{x: |\nabla^k u(x)|> \lambda \}$$ for a function $u\in W^{k,p}(R^d)$ with compact support. We can choose a lower semicontinuous ...
Harcatur's user avatar
4 votes
0 answers
321 views

Topological field theories and their path integrals

Examples of topological field theories are the cohomological field theories as they were initially defined by Witten [1]. Such examples include the Donaldson-Witten theory in 4d or the Gromo-Witten ...
Gorbz's user avatar
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2 votes
1 answer
232 views

A measure of noncompactness by a convex function

Let $E, \left \| \right \|$ be a Banach space, $\mathfrak{M}_E$ indicate a family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the familly of all relatively compact sets, and $Ker \mu=\{X\...
Motaka's user avatar
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0 votes
1 answer
89 views

Verifying that a map to $L^2_{\text{loc}}$ is continuous

Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
geometricK's user avatar
  • 1,851
2 votes
1 answer
303 views

Differentiation on $[0,1]$

EDIT: Perhaps a more reasonable question after thinking about the answer I got would have been. Is there a set $N$ of measure $1-\varepsilon$ and a disjoint partition of that set $N$ with finitely ...
Sascha's user avatar
  • 506
1 vote
0 answers
114 views

Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$

Given a finite set of moment values $\{\mu_1,\ldots,\mu_N\}$, for which the multi-dimensional finite Hausdorff moment problem is determined. That is, we know that at least one distribution $\mathcal{D}...
wzell's user avatar
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1 vote
1 answer
163 views

Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?

Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$ Moreover, we know ...
Sascha's user avatar
  • 506
2 votes
1 answer
102 views

Measurability of the product on particular topological vector spaces

Let $X$ be a topological vector space. Let us say that $X$ has property P if there exists a sequence of closed subsets $\{X_n\}$ such that 1- $X=\bigcup X_n$ 2- The relative topology is both ...
ABB's user avatar
  • 3,898
2 votes
0 answers
39 views

Distribution of a multivariate continuous process determined by that of linear combination of its coordinates?

To keep the question short: Let $C([0,1], \mathbb{R}^d)$, $d \geq 2$ be the space of all $\mathbb{R}^d$-valued continuous processes. $X$ and $Y$ are two $C([0,1],\mathbb{R}^d)$-valued random variates, ...
Dormire's user avatar
  • 223
1 vote
1 answer
298 views

Bochner measurable; continuous operator

It is well-known that if there is a function $f: \Omega \subset \mathbb R^n \rightarrow X$ with $\Omega$ open and $X$ is a Hilbert space, then continuity of $f$ implies also Bochner measurability of $...
Sascha's user avatar
  • 506
6 votes
1 answer
3k views

Measurable functions with non measurable image

I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable. This is motivated by the question Is measure preserving function almost surjective?, ...
user39115's user avatar
  • 1,785
0 votes
1 answer
133 views

Change of variables for double integral [closed]

Thank you for your time. My basic question is whether the following change of variables allowed $$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$ I fail to ...
Xing Wang's user avatar
3 votes
0 answers
125 views

Does every non-locally compact metric space admit a violation of Lebesgue's theorem?

From the results of Preiss and Tišer, it is known that many natural families of measures on Hilbert spaces violate the Lebesgue Density Theorem. Question: Does every non-locally compact metric space ...
Aryeh Kontorovich's user avatar
1 vote
0 answers
1k views

Sigma algebra of stochastic process

A stochastic process is a collection $(X_t)_{t\in T}$ of random variables from a prob. space $(\Omega,\mathcal{F},P)$ to some measurable space $(E,\mathcal{E})$. Now, in order to understand the whole ...
aaaaaaaa's user avatar
1 vote
0 answers
137 views

Weak$^*$ topology on Bochner $L^p$-spaces

Assume that $X$ is a Banach space such that the dual $X'$ has the Radon-Nikodym property. Moreover let $(\Omega,\Sigma,\mu)$ a say finite measure space. Then we know that for $1\leq p<\infty$ holds ...
Miguel Chapman's user avatar
2 votes
1 answer
555 views

Duality of Bochner $L^{\infty}$ space

Let's have a look to the unit interval $[0,1]$ and a Banach space $X$ and then to the space $$ E:=L^{\infty}([0,1],X), $$ i.e. all essentially bounded Banach-valued functions $f:[0,1]\rightarrow X$. ...
Miguel Chapman's user avatar
4 votes
2 answers
413 views

Non-measurability of time integral of non-jointly measurable process

I'm teaching a seminar on probability theory and I want to motivate why joint measurability of a stochastic process is important. The following seems to be the canonical counterexample for a process ...
S.Surace's user avatar
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3 votes
1 answer
288 views

Measurably-isomorphic subsets of polish spaces and the continuum hypothesis

In Theorem 2.7 in the following notes, we seem to assume the following statement. Let $(\Omega,\mathcal F)$ be a Polish space, and $A\in\mathcal F$ an uncountable set. Then there exists a bijection ...
Dominic Wynter's user avatar
2 votes
1 answer
229 views

Absolute continuity of infinite product of probability measures

Let $(A_i,\mathcal{B}_i,\mu_i)$ for $i=1,2,\ldots$ be a sequence of probability spaces. Let $\nu_i$ be another sequence of probability measures on the same underlying measurable spaces. Assume that $\...
Mingchen Xia's user avatar
3 votes
1 answer
202 views

A particular measure of noncompactness?

I am working on an article based mainly on the notion of Measure of non-compactness, to study a particular type of fixed point theorems. Let $\mathcal M $ to be the family of all nonempty bounded ...
Motaka's user avatar
  • 291
9 votes
2 answers
735 views

Functions that are approximately differentiable a.e

The classical definition of an approximately differentiable function is as follows: Definition. Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
Piotr Hajlasz's user avatar
3 votes
2 answers
259 views

The disintegration of the convolution of two probability measures

Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
Alex M.'s user avatar
  • 5,207
3 votes
1 answer
77 views

Convergence of some object depending on functions with compact support

Let $G$ be a locally compact group with unimodular Haar measure $\mu$. We consider the Hilbert space $\mathscr{H}:= L_{\mu}^2(G)$ together with the unitary representation $\pi : G \to U(\mathscr{H})$ ...
Constantin K's user avatar
16 votes
2 answers
972 views

Is measure preserving function almost surjective?

Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one? This question is motivated by the following observation. If ...
Zuofeng Shang's user avatar
0 votes
1 answer
154 views

Positive upper asymptotic density and equidistribution

Let $B=\{b_n: n\geq 1\}$ be a set of positive integer numbers with positive upper asymptotic density and let $\alpha$ be a real irrational number. Is it true that $\{b_n \alpha\}$ is equidistributed ...
Jean's user avatar
  • 515
2 votes
1 answer
192 views

Non-uniqueness in Krylov-Bogoliubov theorem

So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$. Of course, if $X$ is just a ...
Bjørn Kjos-Hanssen's user avatar
1 vote
0 answers
66 views

Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$

Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
PepitoPerez's user avatar
2 votes
2 answers
158 views

Identity map between $L^2(\mu)$ and $L^2(\mu_{\rm sf})$

Let $(X,\Sigma,\mu)$ be a measure space. The semi-finite version of $\mu$ on $(X,\Sigma)$ is denoted $\mu_{\rm sf}$ and given by $$ \mu_{\rm sf}(E) = \sup\{\mu(A) \mid A \subseteq E \text{ measurable,...
Student's user avatar
  • 1,154
2 votes
1 answer
2k views

Explicitly representing a random variable in terms of indicator functions

Motivation: I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula. I want to prove the change of variable formula (you ...
BCLC's user avatar
  • 237
1 vote
1 answer
192 views

Is this function measurable?

Let $(B, \Sigma_B)$ and $(C, \Sigma_C)$ be standard Borel spaces and let $\mu$ be a sub-probability measure on $C$. Given $Y\in \Sigma_{B\times C}$, I would like to use the following function: $$ f:...
daon's user avatar
  • 239
1 vote
1 answer
66 views

Measurability of kernel on generating set

Suppose $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ are measurable spaces such that $\Sigma_Y$ is generated by a set $B$. Suppose $k : X \times \Sigma_Y \to [0, 1]$ has the property that $k(x, -)$ is a (...
daon's user avatar
  • 239
4 votes
2 answers
141 views

Invariant function for Koopman operator of measure-class preserving tranformation

Let $(X,\mu)$ be a standard probability space and let $T:X \to X$ be a measure-class preserving transformation such that there is no $T$-invariant measure absolutely continuous with respect to $\mu$. (...
burtonpeterj's user avatar
  • 1,689
0 votes
1 answer
121 views

Approximation of a measure on $\mathbb{R}^d$

Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix: $$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$ I'm trying to prove the existence of a probability measure ...
user avatar
7 votes
2 answers
851 views

If a measure $\mu$ and Lebesgue measure $\lambda$ are singular, is the derivative of $\mu$ with respect to $\lambda$ $\infty$, $\mu$-a.e.?

If a positive Radon measure $\mu$ and the Lebesgue measure $\lambda$ are singular, can we show that the derivative of $\mu$ with respect to $\lambda$ is $\infty$, $\mu$-a.e.? Namely, can one show that ...
ohliv's user avatar
  • 73
3 votes
0 answers
81 views

Is Lebesgue measure a measurable mapping under Fell topology?

Suppose $\mathcal{F}$ is the space of closed set on $\mathbb{R}$ equipped with the Fell topology. Question: Is Lebesgue measure a Borel measurable mapping from $\mathcal{F}$ to the extended real ...
Uchiha's user avatar
  • 87
2 votes
1 answer
138 views

Measurability of $T \to \Pi_{ker T}$ w.r.t. SOT

I came across the following technical question, to which I could not - after some time of thinking - find an answer: Let $\mathcal{U},\mathcal{H}$ be two real (in general infinite dimensional) ...
PDEprobabilist's user avatar
0 votes
1 answer
52 views

Binarily universal members of $[0,1]$

Let $r\in[0,1]$. We look at the binary represenation of $r$ and say that $r$ is binarily universal if every finite binary string appears in at least one place in the binary representation of $r$. Let $...
Dominic van der Zypen's user avatar
3 votes
1 answer
173 views

A conjecture: Given $nD$ finite measures that bisects

Given $nD$ finite measures $µ_1, . . ., µ_{nD}$ in $\mathbb{R}^n$ does there exists a set of at most D hyperplanes that bisect each of the measures? How far is this problem explained? That is ...
Ma Joad's user avatar
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