**2**

votes

**1**answer

99 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**0**

votes

**0**answers

42 views

### Does a positive-measure subset of the unit interval almost surely intersect a random translation of some countable subgroup of $\mathbb{R}$?

Please forgive me if this is a very easy question.
Let $A \subset [0,1]$ be a Borel-measurable set with strictly positive Lebesgue measure. Does there necessarily exist a countable subgroup $G$ of ...

**4**

votes

**1**answer

182 views

### continuity of the Boltzmann entropy in the Wasserstein metric

For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
...

**0**

votes

**1**answer

66 views

### Approximation of general measurable maps by simple functions [closed]

Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...

**2**

votes

**1**answer

58 views

### Sub-$\sigma$-algebras and conditional expectation

Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space?
In other words, is it true that conditional ...

**5**

votes

**0**answers

140 views

### Sierpinski sets and extensions of Lebesgue measure

I am duplicating an old problem from stackexchange:
Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure ...

**0**

votes

**1**answer

173 views

### Axiomatic explanation of why the volume of a parallelepiped is equal to the area of its base times its height [closed]

I asked this in MSE, it flashed and disappeared.
Let $V_n$ be the volume on the set of polytopes in $\mathbb R^n$, defined axiomatically, i.e. a functional, that assigns to each polytope ...

**0**

votes

**1**answer

34 views

### Convergence of generalized expectation under total variation norm

Consider the space of sub-distributions (i.e. positive measures of variation norm lower than 1) over a discrete subset $S$ of $\mathbb{R}$ (this set is measured by its powerset).
Let $(\mu_n)_n$ be a ...

**4**

votes

**0**answers

217 views

### Is $f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}$ essentially bounded?

Let $$f(x,y)=\sum_{n\in\mathbb{Z}\backslash\{0\}}\frac{1}{n}e^{2\pi i(xn+yn^2)}
$$
Is it true that $\|f\|_{L^{\infty}(\mathbb{R}^2)}<\infty$? i.e. is $f$ essentially bounded?

**1**

vote

**1**answer

51 views

### When is the hitting time of an open set a stopping time?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_t)_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be ...

**1**

vote

**0**answers

75 views

### Any relationship between Hausdorff measures [migrated]

Let $ S_1= ( [0,1], d_1 ) $ and $ S_2 = ( [0,1], d_2 ) $ be two metric spaces, where $ d_1 = |x - y|$ and $d_2 = (1/2^i) $ where binary expansion of x and y matches upto $ i^{th} $ coordinate. Let $ ...

**4**

votes

**1**answer

195 views

### Measure algebra of a total extension of Lebesgue measure

Solovay shows that the existence of a measurable cardinal is equiconsistent with the existence of a countably additive extension of Lebesgue measure that is defined on all sets of real numbers. Given ...

**0**

votes

**0**answers

33 views

### The use of Haar measure in the Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result
Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...

**-3**

votes

**0**answers

47 views

### invariant mesures and entropy

Fix {0,1}^{N} the Bernoulli space, the shift function S and pick two dynamical systems (S,m1) (S,m2) (m1,m2 S-invariant measures). Let's say you have entropies h1 and h2 for the dynamical systems. ...

**4**

votes

**2**answers

182 views

### Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**8**

votes

**1**answer

183 views

### Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...

**0**

votes

**0**answers

114 views

### Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).
...

**1**

vote

**1**answer

69 views

### First mean value theorem for integration and Lebesgue measureability

According to first mean value theorem for integration, if $G \ : \ [a,b] \to \mathbb{R}$ is a continuous function, there exists $x \in (a,b)$ such that
$$\int_a^b G(t) dt = G(x)(b-a)$$
Assume $G$ is ...

**1**

vote

**0**answers

105 views

### Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable).
Also, let $f:D_1\cup ...

**0**

votes

**0**answers

36 views

### Interchange limit with integral for subsequence in subsequence of general distribution functions

I asked this question on math.stackexchange a few days ago but didn't get any response, so I thought I would try here.
I'm trying to find a solution for the following problem:
Let ...

**0**

votes

**2**answers

112 views

### Continuity of an integral

Let $f \in L^1(\Omega)$, where $\Omega$ is a bounded set in $\mathbb{R}^n$
Let $Z = (z_1,z_2,\dots,z_k)$ denote a k-tuple where each $z_i \in \Omega$
Consider $$F(Z) = \int_\Omega f(y)\min_{1\leq i ...

**0**

votes

**0**answers

35 views

### convex function with distributional Hessian $D^2 f \leqslant \lambda$, $\lambda$-concave?

Let $f:R^n \to R$ be convex (may not $C^1$), $$[D^2f]=[D^2f]_{ac}+[D^2f]_s=[h_{ij}] L^n+[D^2f]_s$$ is the Lebesgue decomposition of the Hessian matrix. Where $[h_{ij}]$ is the density w.r.t the ...

**2**

votes

**2**answers

283 views

### Can one estimate the distribution of eigenvalues of a matrix by its Cauchy/Stieltje transform?

Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$
Is there any information ...

**6**

votes

**1**answer

215 views

### Can ITTM recognize a non-measurable set?

Throughout the question ITTM refers to Hamkins' infinite Turing machines, though I will be interested in results related to stronger models.
Recently I was wondering, is it consistent that there is ...

**2**

votes

**0**answers

47 views

### Sequences of transition probability measures

Suppose that $X$ and $Y$ are compact metric spaces. A Borel probability measure $\mu$ on $X\times Y$ satisfies
$$
\mu(A\times B)=\int_A\mu(B|x)\mu_X(dx),
$$
for $A$ and $B$ Borel sets in $X$ and $Y$ ...

**4**

votes

**2**answers

630 views

### Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish?
I have a specific space in mind, so if the ...

**5**

votes

**1**answer

113 views

### Simultaneous approximation of arbitrary functions in Hölder space and in $L^2(\mu)$ by a smooth function and its derivative

Let $\mu$ be a probability measure on the circle $S^1=\mathbb{R}/\mathbb{Z}$ which is singular with respect to the Lebesgue measure $\lambda$. Consider the functions spaces $L^2(\mu)$ on the one hand, ...

**5**

votes

**2**answers

77 views

### Accuracy of the truncated Hausdorff moment problem

For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that
$$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$
In other words, $M_s$ ...

**0**

votes

**1**answer

60 views

### Convergence of measures to an absolutely continuous measure

Suppose that $\{\mu_n\}$ is a sequence of Borel probability measures on a compact metric space $X$ and suppose that $\{\mu_n\}$ converges weakly to a Borel probability measure $\mu$ on $X$. If $\mu$ ...

**7**

votes

**3**answers

266 views

### Borel coloring of a graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$

The following question was asked in a comment by Joel David Hamkins in Graph on the set of all functions $f:\mathbb{N}\to\mathbb{N}$.
Let $V$ be the set of all functions $f:\mathbb{N}\to\mathbb{N}$. ...

**0**

votes

**1**answer

110 views

### topology of setwise convergence of measures

It is well known that if $X$ is, say, compact and metric, then the set of probability measures on the Borel subsets of $X$ endowed with the usual topology of weak convergence of measures has as a ...

**3**

votes

**1**answer

94 views

### Measurability and continuity for general topological spaces

Let $(X,\tau)$ be a topological space. We call $S\subseteq X$ saturated if $S=\bigcap\{U\in\tau: U\supseteq S\}$. Let $\sigma(X,\tau)$ be the $\sigma$-algebra generated by $\tau\cup\{K\subseteq X: K ...

**8**

votes

**1**answer

129 views

### Specifying $L^p$ norms of derivatives

Given a sequence of positive numbers $\{a_n\}$ and $1 < p < \infty$, $p\neq 2$, is it possible to build a function $f\in C^\infty(\mathbb R)$ so that $\|f^{(n)}\|_{L^p(\mathbb R)} = a_n$?
For ...

**0**

votes

**3**answers

142 views

### When Banach indicatrix is measurable?

Let $f:X\to Y$ is a measurable function. Banach indicatrix
$$
N(y,f) = \#\{x\in X \mid f(x) = y\}
$$
is the number of the pre-images of $y$ under $f$. If there are infinitely many pre-images then ...

**6**

votes

**1**answer

279 views

### An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that
$$
f^2=f
$$
(that ...

**7**

votes

**1**answer

288 views

### Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?

In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition.
(EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of ...

**3**

votes

**2**answers

195 views

### When does a function space allow for point evaluations? [closed]

Consider a space of (generalized) functions $F$ defined on a measure space $\Omega$ and equipped with a topology.
What are necessary and sufficient conditions for point evaluations at arbitrary $x ...

**1**

vote

**0**answers

49 views

### Pollard's construction of measures from set functions on lattices of sets

Theorem 12 in Appendix A of Pollard's A User's Guide to Measure Theoretic Probability gives conditions under which a set function defined on a family of sets $\mathscr{K}$ which is closed under finite ...

**-1**

votes

**1**answer

105 views

### Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...

**7**

votes

**0**answers

215 views

### A question about finitely additive extensions of Lebesgue measure

Suppose $m:P([0, 1]) \to [0, 1]$ is a finitely additive measure extending the Lebesgue measure. Must there exist some $X \subseteq [0, 1]$ such that $m(X \cap I) = |I|/2$ for every sub interval $I ...

**4**

votes

**1**answer

136 views

### convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on
a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$.
Suppose I know that
$$\int f d \mu_n \to \int f d\mu$$
...

**2**

votes

**1**answer

72 views

### Quotient sigma-algebra generated by quotient-measurable generating sets

Let $X$ be a measurable space whose $\sigma$-algebra is generated by a family $\mathcal{G}=\bigcup_n \mathcal{G}_n$ of subsets of $X$, where $(\mathcal{G}_n)$ is a sequence of $\sigma$-algebras on $X$ ...

**10**

votes

**1**answer

240 views

### Are there any exact results for Hausdorff Measure?

The computation of the Hausdorff measure is extremely difficult due to the infinum appearing in its definition. This has made the calculation of the Hausdorff measure for nearly all fractals difficult ...

**3**

votes

**2**answers

172 views

### Plane measurable sets and measurable rectangle

Does every measurable set in the plane with positive Lebesgue measure contain a cartesian product of two measurable sets of the real line with positive Lebesgue measures?

**4**

votes

**1**answer

183 views

### Meager set of full measure

Let X be a compact Hausdorff topological group and let m be the Haar measure on X. Can we find a meager set in X whose complement is m-null? I can do it when X is separable but I don't know if there ...

**0**

votes

**0**answers

92 views

### Alternate proof for Caratheodory extension theorem

This question is on the intuition behind the Caratheodory definition of measurable sets as given in Billingsley. He motivates by saying that we "should" call a set $A$ measurable if $$P^*(A) + ...

**2**

votes

**2**answers

172 views

### How to show that there's a continuous function separating convex sets of Radon measures?

First, the setup: $X$ is a compact set. By Riesz's representation theorem $C(X)^*=${all Radon measures on $X$}. $K$ is a convex, closed set of probability measures. $m$ is a probability measure out of ...

**3**

votes

**2**answers

240 views

### Topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$

Are there complete TVS topologies for which $\mathcal{M}(X)\otimes \mathcal{M}(Y)$ is dense in $\mathcal{M}(X\times Y)$
This question is strongly linked to
is the space of all borel measures on ...

**0**

votes

**1**answer

121 views

### Is the following “section-wise” defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form
Proposition: Assume that ...

**11**

votes

**2**answers

285 views

### A measure on the space of probability measures

This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...