**2**

votes

**2**answers

176 views

### Birkhoff Ergodic Theorem or Counterexample

The Birkhoff Ergodic Theorem states:
Let $(X,\mathcal{B},m)$ be a finite or sigma finite measure space. Suppose $T:(X,\mathcal{B},m)\to (X,\mathcal{B},m)$ is measure-preserving and $f\in L^1(m)$. ...

**10**

votes

**1**answer

242 views

### smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...

**-2**

votes

**1**answer

105 views

### a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...

**15**

votes

**3**answers

517 views

### Which distributions can you sample if you can sample a Gaussian?

Explicitly: You have a computer that is able to pick a real number at random according to the normal distribution: $\mathcal{N}(0,1) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$. Which distributions can this ...

**10**

votes

**1**answer

261 views

### Is this property equivalent to Lusin's property (N) for continuous functions?

A function $F:[0,1]\rightarrow\mathbb{R}$ satisfies Lusin's (N) property if for every measure zero set $A\subseteq [0,1]$, $F(A)$ has measure zero. (This includes the assertion that $F(A)$ is ...

**0**

votes

**1**answer

133 views

### Dense subsets on set space

Let $X$ be a metric space, and $\mathscr{B}$ the $\sigma$-algebra generated by open sets of $X$. Can we find a countable dense subsets of the metric space $(\mathscr{B},d)$ with the metric ...

**1**

vote

**1**answer

67 views

### When is $L^{2}(X,\mathscr{B},m)$ spearable [duplicate]

If $X$ is a metric space, $m$ is a Borel probability space on $(X,\mathscr{B})$ where $\mathscr{B}$ is the $\sigma$-algebra generated by open sets on $X$, can we prove that the space ...

**5**

votes

**1**answer

112 views

### Regularity of Patterson-Sullivan Length function

Let $(M,g)$ be a negatively curved, closed Riemannian manifold. I'll ask the question first, then explain the involved players. This data defines the Patterson-Sullivan length function,
...

**5**

votes

**1**answer

243 views

### What it is the volume of the unit ball section of the cone of positive definite matrices?

Let $PD_{n}$ be the cone of positive definite $n \times n$ real matrices and let $B$ be the unit sphere in $n \times n$ dimensions. What is the volume of $PD_{n} \cap B$?
EDIT: Let's assume that $B$ ...

**9**

votes

**0**answers

129 views

### Haar measurable sets and quotient maps

Let $G$ be a locally compact Hausdorff group with a Haar measure $\mu$, let $H$ be a closed normal subgroup of $G$, and let $q: G \to G/H$ be the quotient homomorphism. Let $\nu$ be a Haar measure ...

**0**

votes

**1**answer

89 views

### Is any invariant, ergodic measure with full support on an irreducible Markov shift a Markov measure?

I have this question I have been struggling with for a while. It seems rather intuitive, however, I was not able to proof it yet:
Let $\Omega = \{1,2,\cdots,N\}$ a finite alphabet, $\Sigma \subset ...

**0**

votes

**0**answers

67 views

### Measure generated by Semigroup $\exp[-t|p|]$

I am studying Ingrid Daubechies' paper 'An Uncertainty Principle for Fermions with Generalized
Kinetic Energy'.
On page 514, the measure $\mu_{x,y;t}$ is introduced as generated by the semigroup ...

**2**

votes

**0**answers

40 views

### Measurability of functions with multiple parameters

For a formalisation of the Giry monad in a theorem prover, I think I require some notion of measurability of “curried” functions. I.e. I have measure spaces $A$, $B$, and $C$ and a function $f: A ...

**2**

votes

**1**answer

82 views

### Convex interaction energy

Does anybody know examples of absolutely continuous probability measures $\mu_0,\mu_1$ on $\mathbb{R}^n$ with finite 2nd moments such that
$$
\frac{d^2}{dt^2}\left(\int_{\mathbb{R}^n\times ...

**9**

votes

**2**answers

286 views

### Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...

**7**

votes

**1**answer

203 views

### Absolute continuity reflected in Fourier coefficients?

Imagine $\mu$ and $\nu$ are two Borel probabilty measures in the interval $[0,1]$.
We say that $\mu$ is absolutely continuous with respect to $\nu$, if for every measurable set $A$ such that ...

**3**

votes

**0**answers

104 views

### $f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...

**13**

votes

**0**answers

298 views

### A question about small sets of reals

In ZFC, does there exist an uncountable set of reals $A$ such that for every closed measure zero set of reals $B$, $ A + B = \{a+b : a \in A, b \in B\} \neq \mathbb{R}$?
This question is motivated ...

**1**

vote

**1**answer

161 views

### question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g
...

**4**

votes

**1**answer

156 views

### Homeomorphisms that admit a decomposition

Let $I^n:=[0,1]^n$ and $T$ be a homeomorphism on $I^n$.
If $T$ admits a decomposition of $I^n=A\cup B\cup C$ with $A,B,C$ Lebesgue measurable and mutually disjoint such that $$T(A)=B, T(B)=A \ ...

**3**

votes

**1**answer

129 views

### Multivariate monotonic function

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ ...

**-1**

votes

**2**answers

119 views

### conditional expectation under convex combinaison of probability measures(II)

Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...

**1**

vote

**2**answers

157 views

### Empirical estimator for total variation distance between two product distributions

Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not ...

**4**

votes

**2**answers

173 views

### Tightness of Measures, Riesz Representation for locally compact spaces

Let $X$ be a locally compact, separable metric space and $\mu_n$ a sequence of probability measures on $S$. Let $\mathfrak{C}$ be a convergence determining class for the weak topology (for instance, ...

**4**

votes

**1**answer

314 views

### Lebesgue measure of some set of irrational numbers

Let $(i_{n})$ be a strictly increasing sequence of natural numbers,
$(v_{n})$ be an unbounded sequences of natural numbers
and $M\geq 2$. Denote by $\mathcal{I}(i_{n}, v_{n}, M)$ the set of all ...

**3**

votes

**0**answers

165 views

### Do all $L^{\infty}(\mu)$ spaces have the Grothendieck property?

Consider $L_{\infty}(\Omega,\Sigma,\mu)$, where $(\Omega,\Sigma,\mu)$ is any measure space. Does it it have the Grothendieck property? If the measure space is localizable, then it is true. The ...

**3**

votes

**1**answer

139 views

### conditional expectation under convex combinaison of probability measures

Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...

**1**

vote

**1**answer

62 views

### Measures of disjoint unions and complements of a collection of sets

Let $\mu$ be a probability measure. Let $\mathcal A$ be a collection of measurable sets and $D(\mathcal A)$ be the minimal $\lambda$-system (Dynkin system) containing $\mathcal A$.
Is $\mu(D)$ for ...

**2**

votes

**0**answers

76 views

### Existence of a conditional distribution

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...

**1**

vote

**0**answers

43 views

### Product of sigma-subadditive functions

Let us call a function $\mu : \mathcal{H} \to [0, \infty]$ $\sigma$-subadditive, if $\mu(A) \le \sum_{i \in I} \mu(A_i)$ for every $A \in \mathcal{H}$ and for every countable family $(A_i)_{i \in I}$ ...

**4**

votes

**3**answers

149 views

### Is the range of a (nonnegative or signed) measure a closed set?

Halmos showed that the range of a non-negative, finite measure is a closed subset of real numbers.
Is this true for non-negative, even infinite measures?
Is this true for signed measures? If so, can ...

**4**

votes

**0**answers

157 views

### Conditional expectation with respect to random closed sets

Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...

**4**

votes

**1**answer

101 views

### General version of Skorokhod representation of random variables

Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the ...

**3**

votes

**2**answers

330 views

### Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...

**5**

votes

**1**answer

76 views

### Measurability of $\{ x \in X ; H_0 x \subset A \}$

Let $H$ be some Polish group and $X$ some standard Borel space. Assume that $H$ acts measurably on $X$, i.e. $(h,x) \mapsto hx$ is Borel. Let $H_0 \subset H$ and $A \subset X$ be some Borel sets. Is ...

**2**

votes

**1**answer

156 views

### Is this integration by parts legitimate?

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$
$$
f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,
$$
which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...

**2**

votes

**1**answer

149 views

### reference request: Riesz/Newton potential and HLS inequality in L1.logL1

Let's consider in dimension $d\geq 3$ the Newton/riesz potential $f=I_2[g]$
$$
f(x)=\int_{R^d}\frac{1}{|x-y|^{d-2}}g(y)dy,
$$
which solves $-\Delta f=g$ (up to positive normalizing constants, which I ...

**2**

votes

**1**answer

97 views

### Pullback of $L^p$ functions via exponential map

Let $M$ be a complete Riemannian manifold, endowed with its exponential map $\exp: TM \longrightarrow M$. For any $C^k$- function $u$, we get the Pullback
$$ \exp^* u = u \circ \exp$$
which is in ...

**2**

votes

**2**answers

126 views

### Integral over integrals with different measures

I was trying to construct a category of measurable spaces and 'nondeterministic functions' (not the usual category of measurable spaces); specifically a map $(\Omega, \Sigma)\rightarrow (\Omega', ...

**9**

votes

**1**answer

222 views

### Is the “continuous on compact subsets” characterization of measurable functions actually useful?

According to Lusin's theorem (and the slightly weaker converse of that result), measurable functions on locally compact topological spaces that are equipped with a regular measure may be characterized ...

**3**

votes

**2**answers

204 views

### Non-Polish Lebesgue probability space?

Any Lebesgue probability space is mod. 0 isomorphic to some Polish probability space (with $\sigma$-algebra the completion of Borel algebra, and some Borel probability). I would like to see an example ...

**4**

votes

**1**answer

153 views

### Can Haar measure fail to be bi-invariant without conjugation shrinking a set?

(This is a slightly reformatted and clarified version of my question from math.SE, since I believe
the answer there is wrong and its poster has not responded to my comment in over two weeks.)
Let ...

**5**

votes

**3**answers

242 views

### Does every separated measurable space embed into a power of $\{0,1\}$?

Let $(X,\Sigma)$ be a measurable space of arbitrary cardinality. I would like to understand under which conditions this space is isomorphic to a measurable subset of $\{0,1\}^\kappa$ for some cardinal ...

**3**

votes

**1**answer

106 views

### Reference to definition of matrix log with domain SO(3) which is Borel measurable

I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...

**1**

vote

**1**answer

92 views

### Two equivalent measures on the real Grassmannians?

Denote by $G(n,k)$ the real Grassmannian, the set of $k$-dimensional subspaces of $\mathbb{R}^n$. It is a topological space, even metrizable (see A metric for Grassmannians), and so it is a ...

**1**

vote

**0**answers

79 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

**2**

votes

**1**answer

150 views

### The Universal Algebra of a sigma-Algebra

I am searching for the 'dual' algebraic structure of a Sigma Algebra. The notion of duallity is like on the case of the Boolean Algebra and Set Algera.
If X is a set, the complement and intersection ...

**0**

votes

**1**answer

82 views

### Constructing measures with support in a given set

I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...

**5**

votes

**2**answers

226 views

### Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...

**4**

votes

**1**answer

160 views

### What does the set of cardinals admitting a k-additive measure look like?

Consider an infinite cardinal $\kappa$. Is it the case that the existence of a $\kappa$-additive measure on some infinite set implies the existence of such a measure on every infinite set of size ...