Questions about abstract measure and integral theory. Also concerns such properties as measurability of maps and sets.
3
votes
1answer
43 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
173 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
61 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 ...
0
votes
0answers
39 views
Bound on measure of set difference by means of Laplacian [on hold]
I am considering the following setting: Let $f,g$ be sufficient 'regular' probability density function, and $A=\{\Delta f<0\}$ and $B=\{\Delta g<0\}$, what I want is that if $||f-g||<\delta$ ...
1
vote
1answer
126 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 ...
1
vote
1answer
50 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
85 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
116 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
185 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
167 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
1answer
137 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
204 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
102 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
78 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
71 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
118 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
72 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
190 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
136 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
124 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
0answers
41 views
How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?
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
232 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 ...
3
votes
0answers
31 views
How large are measurable cardinals of higher orders? [migrated]
For each ordinal $\alpha$ define the notions of $\alpha$ - measurable cardinals and $\alpha$ - normal measures as follows:
A measure $\mu$ on a measurable cardinal $\kappa$ is a $0$-normal measure ...
2
votes
1answer
101 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
182 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
153 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
42 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
98 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}:~ ...
2
votes
1answer
196 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
83 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
31 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
109 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
52 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
112 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}$ ...
0
votes
1answer
54 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 ...
0
votes
0answers
74 views
Convergence of functions of bounded variation
Let $u_n$ be a sequence of smooth functions with $||u_n||_{W^{1,1}}<C<\infty$. By $BV$ compactness, a subsequence of $u_n$ converges to some $u$ in $L^{n/(n-1)}$. Assume $\int_{\Omega} (|\nabla ...
1
vote
0answers
63 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
147 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
38 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
2answers
142 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
66 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 ...
2
votes
0answers
91 views
Radon-Nikodym derivatives as limits of ratios
Let $\mu_1$ and $\mu_2$ be measures with $\mu_1 \ll \mu_2$. Suppose we can characterize (a version of) their Radon-Nikodym derivative this way:
$$\frac{d\mu_1}{d\mu_2}(x) = \lim_{n \to \infty} ...
2
votes
1answer
130 views
A question about stochastic kernels and invariant measures
Suppose that $E$ is a metric space, let $\mathcal{B}_E$ denote the set of its Borel subsets and suppose that $\mu$ is a probability measure on $(E,\mathcal{B}_E)$. In addition, suppose that $p:E\times ...
1
vote
1answer
126 views
Equivalent Definition of Measurable Function
Suppose $f: \mathbb{R} \to \mathbb{R} $ is a function. Is it equivalent that:
1) $f$ is measurable
2) the area under $f$ (i.e $\{ (x,y)\ | \ f(x)\leq y\} $) is measurable in the product measure of ...
6
votes
2answers
174 views
Which Sigma-Ideals in a Sigma-Algebra are Ideals of Null Sets?
My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
3
votes
1answer
88 views
Generalization of Lévy's continuity theorem for nuclear spaces
I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions ...
1
vote
0answers
115 views
Question on the Quotient Integral Formula
I have a very concrete question on the proof of the following (see below): Given a 'nice' top. space $X$ and a 'nice' group operation of $G$, say, from the right, on $X$ and a certain measure on $X$, ...
2
votes
1answer
116 views
Function from a compact metric space to the subsets of the naturals
Let $X$ be a compact metric space, and $\mu$ a Borel probability measure. For
$S\subset\mathbb{N}$ we denote the upper density with $\overline{D}(S).$
Let $f:X\rightarrow2^{\mathbb{N}}$ be a ...
4
votes
2answers
165 views
Integration on Compact Semirings
I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...
12
votes
7answers
633 views
Examples of toposes for analysts
I've read that toposes are extremely important in modern mathematics, but I find the definitions and examples given on the nLab page a little too abstract to understand.
Can you provide some examples ...
