All Questions
Tagged with pr.probability measure-theory
823 questions
3
votes
1
answer
966
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 ...
- 8,512
1
vote
2
answers
974
views
Existence of limit measure
Let $X$ be a separable metric space, $\mu_{n}$ a sequence of Borel probability measures
and $\mathcal{C}$ be a family of sets that is closed under finite unions and
interections, and that contains all ...
- 695
6
votes
2
answers
793
views
Obtaining conditional probabilities as pushforwards of [0,1]
It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable ...
4
votes
1
answer
636
views
Product of two sigma fields
Let $\mathcal{F}$ and $\mathcal{G}$ be any two famillies of subset of a space $X$ (neither $\mathcal{F}$, nor $\mathcal{G}$ is a sigma-field).
$$\sigma( A\times B , A\in \mathcal{F}, B\in \mathcal{G})...
9
votes
1
answer
357
views
Random variables invariant under almost automorphisms.
Let $\Omega$ be a standard atomless probability space, we can assume $\Omega=(0,1)$ with Lebesgue measure. A bijection $f:\Omega/A_1\to\Omega/A_2$ is almost automorphism, if $P(A_1)=P(A_2)=0$, $f(A)$ ...
- 1,354
5
votes
1
answer
403
views
Is every bornological space measurable?
Every topological space is measurable, since we may canonically equip a topological space with its Borel $\sigma$-algebra. A bornological space is like a topological space, except the structure ...
- 8,071
0
votes
1
answer
229
views
Weak convergence in measure for negligible sets.
Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and ...
- 17k
10
votes
1
answer
1k
views
Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 41.1k
7
votes
2
answers
417
views
Does every commutative monoid admit a translation-invariant measure?
Let $T$ be a commutative monoid, written additively. The set $T$ is equipped with a canonical pre-order, defined by $s \le t$ when there exists $s' \in T$ so that $s + s' = t$. Consequently, $T$ may ...
- 38.6k
0
votes
0
answers
656
views
Extension of probability measure from a finite algebra to sigma-algebra with countable many generators
I apologize for probably trivial question, I am far from this field.
If $\mathcal A$ is a $\sigma$-algebra of subsets of $X$ (for example Borel sets of Cantor space $2^\omega$), can I extend to $\...
- 1
8
votes
4
answers
1k
views
Is a measurable homomorphism on a Lie group smooth?
Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
Edit: My original question said "measurable ...
- 8,512
22
votes
2
answers
2k
views
Can one view the Independent Product in Probability categorially?
One can construct a category of probability spaces, but this category has no products. Now probability theory relies strongly on the ability to build independent products, the product measure. In a ...
CommunityBot
- 1
6
votes
2
answers
552
views
Is there a good concept of a measurable fibration?
In probability theory, there are many results which are valid in purely measurable settings, usually beginning with the assumption, "let $(\Omega, \mathcal F, \mathbb P)$ be an abstract probability ...
- 3,069
2
votes
1
answer
284
views
Coupling of vectors
Let $X = (X_1,X_2)$ and $\hat X = (\hat X_1,\hat X_2)$ be two random variables where $X_i,\hat X_i$ are taking values over the Polish space $E_i$ endowed with their Borel $\sigma$-algebras, where $i=1,...
- 892
3
votes
3
answers
379
views
Support of an infinitely divisible measure.
Hello,
if $G$ is a compact Lie group. Let $\mu$ be an infinitely divisible measure on $G$, such that $e$, the neutral element of $G$, is in the support of $\mu$. Is that true that the support of $\...
- 2,201
4
votes
1
answer
404
views
Streamlined probability measure for tossing infinitely many coins
The standard probability measure over countably many independent coin tosses (i.e., the probability that you get a prescribed prefix of length $v$ is $2^{-v}$) is usually obtained via results in ...
- 41.1k
9
votes
2
answers
586
views
Fixed objects of the M endofunctor on category Meas
Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces.
As Gerald Edgar &...
CommunityBot
- 1
3
votes
0
answers
133
views
What distribution(s) of delays make(s) timing attacks hardest?
$H$ is (Shannon) entropy.
In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$...
CommunityBot
- 1
1
vote
2
answers
520
views
inequality for coupling of measures
Let $X = \prod _{s \in S} \Omega_s$, with $\Omega_s$ finite and all the same, $S$ countable. Let $\mu_1$ and $\mu_2$ be two probability measures on the product space (not necessarily the product ...
- 882
0
votes
0
answers
98
views
coupling of projections and projection of the coupling
Let $C$ be a coupling between two measures, $C= \mu^1 \mbox{ } t \mbox{ } \mu^2$ ($t$ is the symbol of binary operator of the coupling (I can't find a more proper symbol here)). The measures are both ...
- 295
6
votes
1
answer
443
views
Algorithm for numerically approximating the Prokhorov metric?
Question: What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures?
Recall that the Prokhorov distance metrizes the topology of weak(-*) ...
CommunityBot
- 1
1
vote
1
answer
142
views
Linear Maps between $L^1$-spaces of singular measures
I posted the following question also here, but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, ...
CommunityBot
- 1
2
votes
1
answer
469
views
If two probability distributions have the same weak limit and one of them satisfies Large Deviation Principle, what can we say about the other?
If the probability distribution function of two sequences of random variables have the same weak limit and one of the sequences satisfies a Large deviation principle, then does it imply that the other ...
- 586
5
votes
1
answer
437
views
Stationary, ergodic measures from the structuralist point of view
Stationary, ergodic measures are a class of objects very familiar to probabilists. In a sense, these are the weakest generalization of the classic case of independent, identically distributed random ...
CommunityBot
- 1
4
votes
0
answers
233
views
Convergence of probability measures on a generating field of a sigma-field
Let $(\Omega,\mathcal{B})$ be a measurable space and let $\mathcal{F}$ be a generating field of $\mathcal{B}$. Assume $\mathcal{F}$ is standard, i.e. it is countable, and any normalized, non-negative, ...
- 51
3
votes
1
answer
599
views
Is positive part of the kernel measurable?
Let $(E,\mathscr E)$ be a measurable space and $Q:E\times \mathscr E\to\Bbb [-1,1]$ be a signed bounded kernel, i.e. $Q_x(\cdot)$ is a finite measure on $(E,\mathscr E)$ for any $x\in E$ and $x\mapsto ...
CommunityBot
- 1
2
votes
2
answers
571
views
Family of Brownian Motions
I am trying to show the following statement
Let $D\subset \mathbb{R}^2$ be an open and bounded subset. $\Pi=(P^x : x \in D )$ a Family of standard Brownian Motions started at $x \in D$. Then $\Pi$ ...
- 279
4
votes
2
answers
427
views
Choice of predictable (or jointly measurable) eigenvalues and eigenvectors of nuclear-operator-valued stochastic process
Let $q^{ij}$, $i,j\in\mathbb{N}$, be predictable real-valued stochastic processes. Let $(e^i)$, $i\in\mathbb{N}$ be an ONB of a separable Hilbert space $H$. Assume that $Q=\sum_{i,j=1}^\infty q^{ij}...
CommunityBot
- 1
7
votes
1
answer
579
views
Random Functions and Transition Probabilities
Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces. A transition probability from $S$ to $T$ is a function $\pi:S\times\mathcal{T}\to [0,1]$ such that $\pi(s,\cdot)$ is a probability ...
CommunityBot
- 1
48
votes
7
answers
12k
views
What's the use of a complete measure?
A complete measure space is one in which any subset of a measure-zero set is measurable.
For what reasons would I want a complete measure space? The only reason I can think of is in the context of ...
- 136
9
votes
1
answer
405
views
Applied Problems in Probability which can not be modelled on Polish spaces
Probabilist often work on Polish spaces. Does somebody know an ("non-exotic") example, for which it is not possible to work on a Polish space, but instead one has to work on a general measurable space?...
CommunityBot
- 1
6
votes
0
answers
301
views
Generating stationary, ergodic random fields on a homogeneous space
Consider a homogeneous space $M$, which for the sake of concreteness, let's take to be $M = \mathbb R^d$. Fix some space $A$, and consider the space of functions $X = C(M,A)$, along with its Borel $\...
- 8,512
1
vote
1
answer
166
views
Is the following statement true? $E[\xi U^{'}(\xi)] < +\infty$?
I encounter the following problem today. It seems a simple question.
Let $U$ be a real function from $R^+\rightarrow \bar{R}$ satisfying the following conditions:
(1) $U$ is concave, continuous, ...
- 8,559
0
votes
1
answer
329
views
Is it known that every PDF continuous in all $R^n$ has a maximum? [closed]
I'm working with maximum a posteriori estimation and managed to show that every probability density function that is continuous in all $R^n$ always has at least one global maximum. I've search around ...
- 31.5k
3
votes
2
answers
513
views
Sample from a delta-ball in the orthogonal group O(n)
An answer to another question derived a formula for the volume of a delta-ball in $O(n)$. I am wondering if there is a (constructive) way to draw samples uniformly at random from such a region.
For ...
CommunityBot
- 1
4
votes
2
answers
515
views
Measures that satisfy a 0/1 law
The setting is measure on $2^\omega$. That product (independent) measures obey a 0/1 law, i.e, that measurable tail sets all have measure 0 or 1, is well known. I've made some progress extending this ...
- 2,044
7
votes
2
answers
1k
views
Sections measure zero imply set is measure zero?
I have a subset $B\subset\mathbb{R}^n\times\mathbb{R}^m$ that I want to show has measure zero. I know that the sections $B^x = \{y : (x,y)\in B\}$ all have measure zero. I do not know if $B$ is ...
- 16.2k
1
vote
3
answers
1k
views
What does it mean to say "almost always" ?
I have a set, $A$, of $m \times n$ matrices with certain properties and a subset $B$ of $A$. I would like to say that when randomly selecting such a matrix, I am "almost always" never in $B$. I can ...
- 56.9k
7
votes
3
answers
995
views
Kolmogorov probability axioms without non-negativity condition
What is a minimal consistent modification of probability axioms to include negative values?
Is it enough to use a minimal modification of axioms obtained by
formal exclusion of non-negativity ...
- 19
5
votes
0
answers
200
views
Diffusion processes in wide generality
It is common knowledge among schoolchildren that one may define jump diffusion processes in wide generality.
Hard question: What are the most general structures on which one may define something ...
CommunityBot
- 1
2
votes
0
answers
140
views
Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
CommunityBot
- 1
8
votes
0
answers
729
views
Density of countably additive measure in the set of all finitely additive measures.
Let $S$ be a countable discrete set, the following two results are quite easy to prove:
Every countably additive probability measure $\mu$ on $S$ commutes (in Fubini's sense) with every finitely ...
- 6,034
2
votes
2
answers
655
views
Measure on $\omega_1$
Let $\mathcal{O}$ be the $\sigma$-algebra on $\omega_1$ generated by its countable subsets. Is there a ($\sigma$-additive) probability measure on $\mathcal{O}$ that is not concentrated on a countable ...
- 35.5k
17
votes
1
answer
9k
views
Intuitive understanding of the Stieltjes transform
I have been using random matrix theory in signal processing and have some trouble understanding what the Stieltjes transform does.
The gist of my work is that I have an $N\times N$ true covariance ...
CommunityBot
- 1
4
votes
1
answer
219
views
Transfer independance from $\mathbb{N}$ to $\mathbb{N}^2$
Hello
Here is a little problem for which I have no clue, and I don't even know if it is difficult.
Does there exist a measurable (!) function $\psi:[0,1]^2\mapsto [0,1]$ such that if $(X_i)_i$ is a ...
- 8,501
1
vote
1
answer
509
views
Proving Uniform Convergence from AS Convergence
I'm working the proof of the Stone-Weierstrass Approximation theorem using probability theory from "A Second Course in Probability" by Ross and Pekoz. The statement of the theorem in the book omits ...
- 11.4k
5
votes
1
answer
1k
views
Regular Conditional Probability given a natural filtration of a stochastic process
OK, this is kind of re-posting, but I think I can clarify the question more, so it's worth a shot.
Consider a real valued process $(X_t)_{t \leq T}$, cadlag on a probability space $(\Omega, (\mathcal{...
- 8,512
15
votes
3
answers
3k
views
Entropy of a measure
Let $\mu$ be a probability measure on a set of $n$ elements and let $p_i$ be the measure of the $i$-th element. Its Shannon entropy is defined by
$$
E(\mu)=-\sum_{i=1}^np_i\log(p_i)
$$
with the ...
- 1,101
3
votes
2
answers
1k
views
Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?
Hello,
As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem.
My question is whether each real-...
- 41.1k
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
CommunityBot
- 1