# Questions tagged [measure-theory]

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

**10**

votes

**1**answer

303 views

### Are all compact subsets of Banach spaces small in a measure-theoretic sense?

Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...

**2**

votes

**1**answer

161 views

### Convergence rate for $L^2$ convergence

Let $f \in L^2(\mathbb R)$ then it is well-known that
$$ \widetilde{f}(x):=\sum_{n \in \mathbb Z} \frac{1}{\varepsilon}\int_{[n\varepsilon,(n+1)\varepsilon]} f(s) \ ds 1_{[n\varepsilon,(n+1)\...

**0**

votes

**0**answers

73 views

### Caratheodory Differential Equations ( Existence of Solution )

I'm working with filippov's book , "Differential Equations with Discontinuous Right Hand Sides ".
Theorem 1) For $ t_{0} \leq t \leq t_{0} + a ~~~,|x-x_{0}| \leq b $ let the function $ f(t,x) $ ...

**2**

votes

**0**answers

93 views

### Sum-sets of sets of positive measure in the Hilbert cube

Problem. Let $\lambda$ be the standard product measure on the Hilbert cube $[-\frac12,\frac12]^\omega$ and $A,B$ be two $\lambda$-positive Borel subsets of $[-\frac12,\frac12]^\omega$.
Is it true ...

**28**

votes

**2**answers

2k views

### On the probability of the truth of the continuum hypothesis

First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega ...

**8**

votes

**2**answers

301 views

### Approximation of the identity by simple functions

Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi_n:X\to X$ (finite range and measurable) with $\lim\phi_n(x)=x$.
Can we concluded $X$ may be written by a ...

**3**

votes

**1**answer

106 views

### Approximation on separable topological space with size $\mathfrak{c}$

Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function.
Q. Is it possible to find a sequence of ...

**4**

votes

**0**answers

109 views

### A point concerning Fremlin's example on Borel sets in non-separable Banach spaces

Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...

**1**

vote

**0**answers

130 views

### 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 ...

**4**

votes

**2**answers

241 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}^\...

**6**

votes

**1**answer

412 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'} ...

**8**

votes

**1**answer

191 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 ...

**2**

votes

**1**answer

200 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).
...

**8**

votes

**2**answers

376 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 ...

**1**

vote

**0**answers

102 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 ...

**9**

votes

**1**answer

287 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 \...

**5**

votes

**2**answers

334 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 ...

**6**

votes

**3**answers

367 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 $X$?

**1**

vote

**1**answer

103 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}{\...

**5**

votes

**2**answers

285 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 ...

**0**

votes

**0**answers

64 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 ...

**4**

votes

**0**answers

183 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 ...

**0**

votes

**0**answers

173 views

### Does $\int_0^\infty f(x+\theta)g(x) \, dx=0\, \forall \theta \in \mathbb{R}$ imply $f=0$ almost everywhere, if $g$ is smooth and strictly positive?

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be an integrable function, and $g:(0,\infty)\rightarrow(0,\infty)$ a smooth, strictly positive function.
If
$$\int_0^\infty f(x+\theta)g(x)\,dx=0\qquad\forall\...

**3**

votes

**1**answer

185 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\...

**0**

votes

**1**answer

61 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 ...

**2**

votes

**1**answer

235 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 ...

**1**

vote

**0**answers

88 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}...

**1**

vote

**1**answer

128 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 ...

**2**

votes

**1**answer

88 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 ...

**2**

votes

**0**answers

29 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, ...

**0**

votes

**1**answer

123 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 $...

**2**

votes

**1**answer

481 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?, ...

**0**

votes

**1**answer

83 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 ...

**3**

votes

**0**answers

115 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 ...

**1**

vote

**0**answers

225 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 ...

**1**

vote

**0**answers

82 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 ...

**2**

votes

**1**answer

139 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 $f:[0,1]\rightarrow X$. ...

**4**

votes

**2**answers

163 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 ...

**3**

votes

**1**answer

203 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 ...

**2**

votes

**1**answer

93 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 $\...

**0**

votes

**0**answers

62 views

### In Wasserstein, what is the relationship between $W_2 (\widehat{\mathbb{P}}_{N},\mathbb{P})$ and $W_2 (\widehat{\mathbb{P}}_{N}^{x},\mathbb{P}^{x})$?

Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem:
Definition:
The $p$-Wasserstein metric $W_{p}(\mu,\nu)$ ...

**4**

votes

**1**answer

152 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
...

**5**

votes

**2**answers

233 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 $E\subset\mathbb{R}^n$. ...

**3**

votes

**2**answers

173 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,...

**3**

votes

**1**answer

56 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 $\pi : G \to U(\mathscr{H})$ ...

**14**

votes

**2**answers

511 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 ...

**0**

votes

**1**answer

89 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 ...

**1**

vote

**1**answer

92 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 ...

**1**

vote

**0**answers

61 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\...

**2**

votes

**2**answers

121 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,...