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

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3
votes
1answer
271 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
2answers
162 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
2answers
243 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
2answers
218 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, ...
5
votes
1answer
452 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 ...
4
votes
0answers
190 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
1answer
149 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
1answer
66 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
0answers
80 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
0answers
57 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
3answers
168 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
0answers
183 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
1answer
133 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
2answers
439 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
1answer
79 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
1answer
176 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
1answer
162 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
1answer
103 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
2answers
133 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
1answer
281 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
2answers
268 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 ...
6
votes
1answer
162 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
3answers
271 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
1answer
111 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
1answer
103 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
0answers
89 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
1answer
175 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
1answer
85 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
2answers
259 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
1answer
167 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 ...
3
votes
1answer
242 views

When is the bound in Riesz-Thorin Interpolation Theorem attained?

Let me recall the statement of Riesz-Thorin theorem (see also http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem). Theorem (Riesz-Thorin): Let $(X,\mu)$ and $(Y,\nu)$ be $\sigma$-finite ...
0
votes
1answer
141 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
5
votes
0answers
300 views

Skorohod theorem (weak convergence) on a discrete setting

I have a question about the application of Skorohod representation theorem. The questions arises in this paper about robust hedging in mathematical finance. It is about the very last equation on page ...
2
votes
1answer
149 views

Cameron-Martin theorem for non-Gaussian measures

Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^* \to X$. Let $\tau_u : X \to X$ ...
3
votes
1answer
219 views

Density of linear functionals in $L^2$

Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals ...
4
votes
1answer
210 views

Uncountable atomless subalgebras of the Boolean algebra of all Jordan measurable sets in [0,1]

Definition: Suppose $\mathcal A$ is the Boolean algebra of all Jordan measurable sets in $I=[0,1]$ (i.e $\mathcal A=\{A\subseteq I: \mu(\partial(A))=0\}$, where $\mu$ is the Lebesgue measure and ...
0
votes
0answers
63 views

Does there exists $f\in A_{\mathbb R}(\mathbb T)$ with $||f||=r$ such that $||e^{if}||=e^{r}$?

Let $\mathbb Z$, the set of integers, be a group with respect to addition and its dual group is the $\mathbb T = \{z\in \mathbb C : |z|=1\}$ , one dimensional torus. Put, $\ell^{1}(\mathbb Z)= ...
1
vote
0answers
108 views

approximation of probability distribution

I have a question: Let $\mu$ be a probability distribution defined on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ satisfying $$\int_{\mathbb{R}}|x|d\mu<+\infty$$ Set $$A_n=\Big\{\frac{i}{n}:~ ...
3
votes
1answer
338 views

Integral wrt probability measure

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
0
votes
1answer
137 views

additive measure on countable algebras

I was wondering, can the following theorem be true for finitely additive measures defined on algebras not $\sigma$-algebras. (Theorem is in Bogachev's Measure Theory Vol I). I was not sure about ...
0
votes
0answers
43 views

absolute continuity of a measure given absolute continuity of conditionals

Situation is the following. We have the two-dimensional torus $X$ and have partition $\xi$ into vertical circles $\{x\} \times S^1$. We are given a measure $\mu$ on $X$ such that the projection ...
0
votes
1answer
141 views

Some convergence similar to weak-$\ast$ convergence on the space of finite measures

I have a question: Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$. Define $TV(x)$ as the total variation ...
2
votes
0answers
120 views

Weak convergence of $\sigma$-finite measures

Let $(E,\mathcal{E})$ be a measure space and let $(m_{n})_{n\in\mathbb{N}} \subset \mathcal{M}_\sigma(E,\mathcal{E})$ be a sequence of $\sigma$-finite measures. I will put my questions and state below ...
0
votes
1answer
190 views

The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group. Question. What would be natural vector space $\mathcal{R}$ ...
1
vote
1answer
252 views

Sufficient conditions for equality of measures related to harmonic functions

In Axler's book "Harmonic function theory" on this link http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Livres/Harmonic%20Function%20theory.pdf on page 112 ...
1
vote
0answers
90 views

equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)

Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$, $$ ...
3
votes
0answers
196 views

Strong measure zero sets and selection principles

A set of reals $X$ is strong measure zero if for any sequence of positive real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
0
votes
0answers
61 views

Maximal isometrically invariant finitely additive extension of Lebesgue measure in dimension $\ge 3$?

Question: For $n\ge 3$, is there a maximal isometrically-invariant finitely additive extension of Lebesgue measure on $\mathbb R^3$? A maximal extension of a measure with some properties (in this ...
0
votes
3answers
297 views

dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
0
votes
1answer
73 views

Existence of a map connecting two marginals of a product measure

Let $X$ and $\bar X$ be two standard Borel spaces, and let $A\subseteq X\times\bar X$ be an analytic subset of the product space. Let $P$ be any probability measure such that $P(A) = 1$, and denote by ...