**1**

vote

**1**answer

219 views

### A set of positive-measure not being a countable union of cylinder sets and zero-measure sets?

Let $(A^\mathbb{N}, \mathcal{B}(A^\mathbb{N}), \mu)$ be a measure space, where $A^\mathbb{N}$ is a set of one-sided sequences over a finite alphabet $A \subset \mathbb{N}$, $\mathcal{B}(A^\mathbb{N})$ ...

**16**

votes

**2**answers

568 views

### A moment problem on $[0,1]$ in which infinitely many moments are equal

Suppose $\mu$ and $\nu$ are two probability measures on $[0,1]$. Let their $n$-th moments be denoted by $\mu_n$ and $\nu_n$, respectively, for $n \in \mathbb{N}$.
If we know that $\mu_n=\nu_n$ for ...

**0**

votes

**0**answers

101 views

### proof of “supermodular function induces measure”

A function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ induces a measure by its finite differences, that is
\begin{align}
\mu((\mathbf{a},\mathbf{b}]) := \Delta_{a_1,b_1}\cdots\Delta_{a_n,b_n} f
...

**2**

votes

**0**answers

177 views

### Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated.
1) It is the ...

**2**

votes

**1**answer

252 views

### Liouville's theorem: How to get an invariant measure?

Liouville's theorem states that the `natural' 2-from is preserved under the Hamiltonian flow. Apparently this leads to an invariant measure $\mu$ as follows
\begin{equation}
d\mu = \frac{d\sigma}{|| ...

**21**

votes

**2**answers

1k views

### Most significant results in motivic integration theory?

I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, ...

**0**

votes

**2**answers

189 views

### Two questions about Convex Sets and Lebesgue Measure

For any positive integer n, let E(n) denote n-dimensional Euclidean Space and let L(n) denote n-dimensional Lebesgue Measure on E(n). Take n=2 for simplicity. There are uncountably many convex subsets ...

**4**

votes

**1**answer

195 views

### Coupling of non-probability/sub-probability measures

A coupling of two probability measures $P,\tilde P$ on a Borel space $X$ is any probability measure on $X^2$ whose one-dimensional marginals are $P$ and $\tilde P$. In particular, for any such ...

**9**

votes

**1**answer

444 views

### Is there a probability theory developed in intuitionistic logic?

Since Boole it is known that probability theory is closely related to logic.
According to the axioms of Kolmogorov, probability theory is formulated with a (normed)
probability measure ...

**6**

votes

**2**answers

117 views

### Vanishing of integral on hemispheres implies vanishing of function?

Consider a function $F$ on the half space $\{(x,y,z)|z>0\}$. If $F$ is analytic, it is straightforward to show that
A) The integral of $F$ over the hemisphere $(x-x_0)^2 + (y-y_0)^2 + z^2 = R^2$ ...

**4**

votes

**0**answers

198 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

**1**

vote

**1**answer

99 views

### On the existence of a signed measure related to the Hausdorff moment problem

This M.O question reminded me of a problem I've seen a while ago and was not able to solve. Any reference about this kind of problem would be nice.
The problem is the following :
Does there exist a ...

**4**

votes

**1**answer

143 views

### On vanishing signed measure

It is quite easy to show that if $\mu$ is positive finite Borel measure on, say $[0,1]$, and for all $n \in \mathbb{N}$
$$\int_{[0,1]} e^{-nx}\mu(dx)=0$$
holds true, then $\mu=0$. Does this still ...

**8**

votes

**0**answers

342 views

### Does this metric have an official name? Lévy metric? Ky Fan metric?

Let $X$ and $Y$ be random variables taking values in a separable metric space $(S,d)$. The metric I have in mind is
$$\rho(X,Y) = \mathbb{E}[\min\{d(X,Y),1\}]$$
if $X$ and $Y$ take values in the a ...

**1**

vote

**0**answers

150 views

### When does a proper Zariski closed set have measure zero with respect to a conditional measure?

Assume we have a probability measure $\mu$ over $\mathbb{R}^d$ that is absolutely continuous with respect to Lebesgue measure.
Given $m$ polynomials $p_1,\ldots,p_{m}\in \mathbb{R}[x_1,\ldots,x_d]$ ...

**4**

votes

**1**answer

128 views

### Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...

**3**

votes

**0**answers

78 views

### Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...

**3**

votes

**1**answer

174 views

### Extending Tarski's Theorem on invariant measures

Tarski's Theorem says that if $G$ acts on $X$ and $E$ is a non-$G$-paradoxical subset of $X$, then there is a finitely additive $G$-invariant measure $\mu:2^X\to[0,\infty]$ with $\mu(E)=1$.
I am ...

**3**

votes

**1**answer

321 views

### A reference for this possibly well-known fact concerning the Kakeya conjecture?

I believe I have read or heard somewhere that the Kakeya conjecture would follow from appropriate lower bounds for the minimal size of a subset of $\{ 1 , \cdots , N\}$ which contains a translate of ...

**1**

vote

**2**answers

341 views

### Reference request: learn measure theory for PDEs

I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...

**0**

votes

**0**answers

138 views

### Is there a translation invariant measure on an infinite dimensional space 'without points'?

This is just a reference request. I thought I'd come across a paper demonstrating that there is a translation invariant measure on an infinite-dimensional space without 'points' whilst browsing the ...

**0**

votes

**1**answer

134 views

### identically distributed random variables and measure-preserving transformations

Let $X$ and $Y$ be identically distributed bounded random variables defined on a probability space $(\Omega,\mathcal{F}, \mathbb{P})$. I want to know if there always exists an invertible ...

**4**

votes

**1**answer

242 views

### Winning sets of full measure (Schmidt's game)

A quick reminder of the definition of Schmidt's game:
Let ${X}$ be a metric space and ${S\subset X}$ be a subset. Let
${0<\alpha,\beta<1}$ be constants. Bob chooses any open ball
...

**1**

vote

**1**answer

183 views

### Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful.
Let $V \subset H ...

**10**

votes

**2**answers

664 views

### Why do we want maps to be measurable (in countably-additive setting)

When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ...

**5**

votes

**0**answers

203 views

### Existence of an universally measurable pullback

Let $X,Y$ and $Z$ be standard Borel spaces:
topological spaces homeomorphic to Borel subsets of complete separable metric spaces.
Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...

**4**

votes

**1**answer

356 views

### Are there uniformly discrete paradoxical sets in $R^3$?

I think there aren't any discrete paradoxical sets in $R^2$ (any isometry that mapped a discrete set into itself would have to either be a glide-reflection, a translation or a rotation by $2\pi/n$, ...

**6**

votes

**2**answers

241 views

### Borel kernel over an analytic set implies existence of a Borel map

Let $X$ and $Y$ be standard Borel spaces, and let $A\subseteq X\times Y$ be an analytic set with a full projection on $X$: that is $\pi_X(A) = X$. Suppose that there exists a Borel-measurable kernel ...

**3**

votes

**1**answer

124 views

### Maps that are a.e. equal have almost the same graphs

Let $X$ and $Y$ be two measurable spaces, and let $p$ be a probability measure on $X\times Y$. Denote by $p_X$ the marginal of $p$ on $X$, that is an image of $p$ under projection on $X$. Consider two ...

**2**

votes

**1**answer

145 views

### Universally measurable map coincides a.e. with a Borel map

Let $X$ be a standard Borel space: that is, a topological space equivalent to a Borel subset of $\Bbb R$. It is known that for any probability measure $p$ on $X$ and any universally measurable set ...

**6**

votes

**1**answer

500 views

### Do Measurable Cardinals Exist? (assuming ZFC)

In Appendix B of his Uniform Central Limit Theorems (1999), Dudley writes:
It is consistent with the usual axioms of set theory (including the axiom f choice) that there are no measurable ...

**3**

votes

**1**answer

220 views

### Conditional probabilities for all pairs of subsets in $\mathbb R^2$?

Parikh and Parnes showed that one can define a isometry-invariant finitely-additive conditional probability for all pairs of subsets of the interval. Can one extend this result to bounded subsets of ...

**4**

votes

**1**answer

220 views

### Norms for complex measures

I'm searching for a state of the art introduction to norms on the space of complex measures (on $\mathbb R^n $, for example, or some compact subset thereof). I'd be interested in inequalities of the ...

**4**

votes

**3**answers

510 views

### What is the easiest way to show that three lines in two dimensional space do not intersect?

I have two similar questions:
1) Let $X$ and $Y$ be two measure spaces. Suppose for every point
$x \in X $ there exists a set $ \mathcal{U}_x \subset Y $ of full
measure in $Y$. Suppose $V \subset ...

**0**

votes

**0**answers

213 views

### Measure induced by function

It is known that an n-increasing, left-continuous function $f$, on $[0,\infty)^n$ induces a unique positive measure $\mu$ on $[0,\infty)^n$. Say if $f$ was 3-increasing on $[0,\infty)^3$ but also ...

**5**

votes

**1**answer

473 views

### Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one.
$I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...

**3**

votes

**1**answer

209 views

### Prohorov's theorem for random elements of Hilbert space: weak convergence

Let $(\Omega,\mathcal{F},P)$ be a probability space and let $(E,\mathcal{E})$ be a separable Hilbert space ($E$) with Borel $\sigma$-algebra $\mathcal{E}$. For concreteness let us set $E=L^{2}[a,b]$ ...

**3**

votes

**1**answer

180 views

### Conditional law as a random measure and convergence of random measures

I'm looking for a reference book or article for the following two facts. In both statements, a Polish space $E$ and an ambient probability space $(\Omega, {\cal A}, \Pr)$ are given, and I consider ...

**12**

votes

**3**answers

610 views

### How do we express measurable spaces using type theory?

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spaces, but to the best ...

**11**

votes

**3**answers

597 views

### What fraction of n-point sets in the unit ball have diameter smaller than 1?

This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...

**1**

vote

**2**answers

184 views

### An almost orthogonality principle for L^p

I recently asked this question on Math StackExchange and someone suggested that it would probably be more suited for Math Overflow. Since it still has not been answered, here it goes:
If two ...

**2**

votes

**1**answer

114 views

### Cutting a subset in many pieces with controlled perimeter

Let me premise that I am in no way an expert in the subject of this question.
Let's say we have a measure space $X$ with measure $m$ and a reasonable notion of perimeter for (nice enough) subsets of ...

**2**

votes

**2**answers

645 views

### The image of a measurable set under a measurable function.

Let $f:X \rightarrow (Y, \mathcal{Y})$ be an abstract function, with $\mathcal{Y}$ a $\sigma$-algebra on $Y$. Endow $X$ with $f^{-1}(\mathcal{Y})$. Is then $f(X)$ a measurable set in $Y$? If not, are ...

**2**

votes

**1**answer

282 views

### Measurability of subspace of set of all functions

Set $X=\mathbb{R}^n$ and let $X^{I}$, the space of maps from the (bounded or unbounded) interval $I$ to $X$, be endowed with the locally convex topology of pointwise convergence.
Is it true that the ...

**2**

votes

**1**answer

300 views

### When is the support of a Radon measure separable?

Let $X$ be a topological space, equipped with its Borel $\sigma$-algebra $\mathcal B(X)$, and let $\mathbb P$ be a Radon probability measure on $(X, \mathcal B(X))$. Recall that the support of the ...

**6**

votes

**1**answer

271 views

### Arithmetic products of Cantor sets.

Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product
$AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are ...

**4**

votes

**0**answers

63 views

### Importance sampling of finite path of stochastic difference equation

Before passing to question, let me briefly recap what's importance sampling of random variables is about. Suppose $\xi$ is a real-valued random variable with density $f$, and let $g:\Bbb R\to \Bbb R$ ...

**8**

votes

**2**answers

287 views

### Supremum of measure of sets of measure less or equal to 1/2.

Let $(X,d)$ be a metric space equipped with a probability measure $\mu$ (defined on the Borel $\sigma$-algebra on the topology induced by the metric $d$). I am interested in the different values that ...

**1**

vote

**3**answers

535 views

### Dirac measures dense in space of measures? [closed]

Let $I$ be a compact interval and $\mathcal{M}(I)$ the space of (signed) Borel measures. We equip it with the weak topology, i.e. a sequence $\mu_n$ converges to zero if and only if
$$ \left|\int_I ...

**0**

votes

**0**answers

82 views

### Need an explanation of this paragraph “Lebesgue Homoeomorphism”

I will just quote a part of one proof in "On uniformly regular topological measure spaces by Babiker: page 781" vol43 No4 Duke Math. J. 1976.
Let $I$ be the unit interval endowed with Lebesgue ...