All Questions
Tagged with pr.probability measure-theory
823 questions
178
votes
8
answers
31k
views
Why do probabilists take random variables to be Borel (and not Lebesgue) measurable?
I've been studying a bit of probability theory lately and noticed that there seems to be a universal agreement that random variables should be defined as Borel measurable functions on the probability ...
- 4,874
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 ...
- 481
45
votes
5
answers
6k
views
Nonstandard analysis in probability theory
I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...
- 1,302
32
votes
4
answers
4k
views
Is a random subset of the real numbers non-measurable? Is the set of measurable sets measurable?
One might say, "a random subset of $\mathbb{R}$ is not Lebesgue measurable" without really thinking about it. But if we unpack the standard definitions of all those terms (and work in ZFC), it's not ...
- 2,200
32
votes
1
answer
4k
views
Do invariant measures maximize the integral?
Update: The negative answer to the following question has been provided by Matthew Daws, who won, but also rejected, the bounty of 100 euro that I set over the question.
Let $\mathcal M(\mathbb Z)$ ...
- 6,034
29
votes
3
answers
3k
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 (normalized)
probability measure $\mbox{...
- 567
26
votes
3
answers
11k
views
L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
- 833
25
votes
6
answers
6k
views
Proof of Krylov-Bogoliubov theorem
Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
- 897
25
votes
6
answers
10k
views
Metrization of weak convergence of signed measures
Edit: Changed from "Hausdorff" to "metric" spaces.
Let $\mathcal{M}(\Omega)$ denote the space of signed regular Borel measures on a compact metric space $\Omega$. By Riesz-Markov, ...
- 12.7k
23
votes
2
answers
7k
views
What is a Gaussian measure?
Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.
Is there a direct ...
- 8,512
23
votes
1
answer
3k
views
Bochner integral of stochastic process = path by path Lebesgue integral?
After some helpful comments, I realized that I had to repost this question in a more systematic way.
On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
- 473
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 ...
21
votes
5
answers
4k
views
Existence of probability measure defined on all subsets
Let $S$ be an uncountable set. Does there exist a probability measure which is defined on all subsets of $S$, with $P({x}) = 0$ for any element $x$ of S ?
If I remove the condition $P({x}) = 0$, then ...
- 581
21
votes
3
answers
6k
views
Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.
What is the point of $\pi$-systems and
$\mathcal{D}$ / Dynkin /
$\lambda$-systems?
I am an analyst in the process of consolidating my measure theory knowledge before moving on to harder/newer ...
- 1,771
21
votes
2
answers
3k
views
A measure on the space of probability measures
This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
- 361
21
votes
4
answers
2k
views
Is every probability space a factor space of the Haar Measure on some group?
Let P be an arbitrary probability space.
I would like to find a compact topological group $G$ so that the Haar probability measure on $G$ admits a measurable map to the probability space $P$.
By a ...
- 5,898
20
votes
6
answers
19k
views
Intuition for Haar measure of random matrix
What is an intuitive way to understand Haar measure as defined for random matrices, say, $N\times N$ orthogonal or unitary matrices?
My understanding for what Haar measure means for $U(1)$ is that it ...
- 1,890
20
votes
1
answer
2k
views
Does every compact metric space have a canonical probability measure?
Edit: Shortly after this post it was rightly pointed out by @AntonPetrunin that the measure $\mu$ may not be unique. @R W then showed how one can construct a metric space where the limiting measure is ...
- 203
19
votes
3
answers
3k
views
Measure induced on [0, 1] by infinite tosses of biased coin
It is well-known that one can get the Lebesgue measure on [0, 1] by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary.
I was ...
- 675
19
votes
3
answers
1k
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 ...
- 922
18
votes
3
answers
2k
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 ...
- 8,512
18
votes
3
answers
1k
views
Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?
Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space?
Here is one natural ...
- 8,512
18
votes
1
answer
452
views
Is defining measures as functionals ever insufficiently general in practice?
Crossposting from Math Stack Exchange, as it has yet to receive any answers there; the original question is here.
The way I learned measures was as set functions on a $\sigma$-algebra with certain ...
- 183
18
votes
4
answers
1k
views
Reference for a strong intermediate value theorem for measures
Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
- 5,407
17
votes
2
answers
1k
views
Is measure preserving function almost surjective?
Let $F:[0,1]\to[0,1]$ be a Lebesgue measure preserving function. Is $F$ almost surjective, i.e., the image of $F$ has interior measure one?
This question is motivated by the following observation. If ...
- 397
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 ...
17
votes
2
answers
860
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 ...
- 173
17
votes
5
answers
3k
views
Conditional probabilities are measurable functions - when are they continuous?
Let $\Omega$ be a Banach space; for the sake of this post, we will take $\Omega = {\mathbb R}^2$, but I am more interested in the infinite dimensional setting. Take $\mathcal F$ to be the Borel $\...
- 8,512
16
votes
2
answers
1k
views
How often two iid variables are close?
Is there a constant $c>0$ such that for $X,Y$ two iid variables supported by $[0,1]$,
$$
\liminf_\epsilon \epsilon^{-1}P(|X-Y|<\epsilon)\geqslant c
$$
I can prove the result if they have a ...
- 1,352
16
votes
4
answers
1k
views
Continuity on a measure one set versus measure one set of points of continuity
In short: If $f$ is continuous on a measure one set, is there a function $g=f$ a.e. such that a.e. point is a point of continuity of $g$?
Now more carefully, with some notation: Suppose $(X, d_X)$ ...
- 2,221
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 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 ...
- 6,034
14
votes
4
answers
4k
views
What are two independent, uniformly distributed random variables on the unit interval?
I have been dabbling in learning basic things about probability theory and (of course) being of the school of abstract nonsense I have tried to understand things in its language. I apologize if this ...
- 7,273
14
votes
1
answer
2k
views
Prokhorov's theorem in non separable metric spaces
Recently, working in some calculations I needed to use the Prokhorov's theorem
about compactness for probability measures. However, a friend warned me that
I had not the hypotesis of separability ...
- 757
14
votes
1
answer
2k
views
surprisingly difficult filtration problem
I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky:
Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
- 473
13
votes
4
answers
5k
views
What is known about the Gaussian measure of the unit ball in a Hilbert Space?
Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ ...
- 617
13
votes
1
answer
3k
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 ...
- 6,287
12
votes
2
answers
3k
views
Does there exist an event independent of a given sigma-algebra?
The following question came up in a discussion with my advisor:
Let $(\Omega, \mathcal F, \mathbb P)$ be a non-trivial probability space, and suppose that $\mathcal G$ is a proper sub-$\sigma$-...
- 8,512
12
votes
3
answers
870
views
Measure theory in nuclear spaces
Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...
- 8,512
12
votes
2
answers
3k
views
The Borel $\sigma$-algebra of the set of probability measures
Let $X$ be a compact metric space and $M(X)$ the set of all Borel probability measures on $X$.
It is know that $M(X)$ is a convex compact metric space endowed with the weak-* topology i.e.
$(\mu_n)_n \...
- 133
12
votes
3
answers
891
views
Looking for at least one beautiful and not too technical result in asymptotic group theory
We have a student seminar devoted to the problems of asymptotic group theory with some connections to ergodic theory and measure theory in general. Each talk concerns one of the problems of this ...
12
votes
1
answer
1k
views
Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces
Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
- 333
11
votes
4
answers
3k
views
When does a probability measure take all values in the unit interval?
Let $\mathbb{P}$ be a probability measure on some probability space $(\Omega,\mathcal{A})$. Are there conditions on the $\sigma$-algebra $\mathcal{A}$ such that for every real number $c\in [0,1]$ we ...
- 313
11
votes
1
answer
995
views
Choosing a relative large density subsequence from a low density sequence
My question is somewhere in the interface of combinatorics, probability, and measure theory. It is quite ad-hoc, and I wonder if there is a counter example.
Consider for example the unit interval $[0,...
- 502
11
votes
1
answer
2k
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 cardinals, ...
- 8,512
11
votes
2
answers
2k
views
Multi-dimensional moment problem
Let $\mu$ be a measure on $\def\r{\mathbb{R}}\r^n$, $1\le n \le \infty$. Given a (finite) multi-index $\bar{i} = (i_1, i_2, \ldots)$, one can define the moment
$$ m_{\bar i} = \int x_i^{i_1} x_2^{i_2}...
- 12.8k
11
votes
3
answers
565
views
Is Stoch enriched in Met?
Let $\mathsf{Stoch}$ denote the Kleisli category of the Giry monad. That is, $\mathsf{Stoch}$ is a category whose objects are measurable spaces and for which a morphism $f\in\mathsf{Stoch}(X,Y)$ is a ...
- 8,659
11
votes
1
answer
500
views
Uncountable families of measurable sets with pairwise positive intersections
Let $(X,\mathcal{B},\mu)$ be an arbitrary finitely additive probability measure space, let $a>0$ and let $(A_i)_{i\in I}$ be an uncountable family of subsets with measure $\geq a$.
Is there an ...
- 10.6k
11
votes
2
answers
3k
views
Good examples of random variables whose image is not a measurable set?
Are their simple/natural examples of real-valued Borel-measurable random variables whose image is not a Borel set? Something that occurs "naturally"?
I am teaching Doob's lemma (for two real-valued ...
- 2,201
11
votes
2
answers
466
views
Defining measures over frames in place of $\sigma$-algebras
Normally, measures and probability spaces are defined over $\sigma$-algebras. I was wondering what would happen if one tries to define it over frames in place of $\sigma$-algebras? Specifically, ...
- 5,502