All Questions
Tagged with measure-theory pr.probability
823 questions
4
votes
2
answers
667
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Convergence (topology) for $\sigma$-finite measures
I'm having much trouble finding literature that addresses the questions which I write below. I was wondering if someone could help me out to understand better, either by providing references or by ...
- 361
3
votes
0
answers
126
views
Other than Brownian motion, when else is it possible to define "normalized weighted infinite dimensional Lebesgue measure"?
In this article Sourav Chatterjee poses the question, how do we define the measure:
$$d\mu(A)=\frac{1}{Z}\exp\left(-\frac{1}{4g^2}S_{YM}(A)\right)dA$$
The $Z$ here is an infinite normalizing ...
user69208
2
votes
1
answer
326
views
What is the Wiener measure of the curves with Hölder index $\frac 1 2$?
One may show that the Wiener measure (for curves in $\mathbb R^n$) is concentrated on the Hölder-continuous curves of Hölder index $< \frac 1 2$. What happens to the curves of Hölder index ...
- 5,407
6
votes
1
answer
661
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On the failure of extending a probability measure on uncountable $\Omega$
It is a well known fact that if $(\Omega, \mathcal{F}, P)$ is a probability triple and $\{A_i : i < k\}$ is a finite collection subsets of $\Omega$, then there is a $P' \supset P$ and $\mathcal{F'} ...
- 1,328
6
votes
2
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502
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Continuity of disintegrations
Suppose that $\pi:Y\to X$ is a continuous surjection from one compact metric space to another. Given a regular probability measure $\mu$ on $Y$ with pushforward measure $\nu:=\pi^*\mu$, it is known ...
- 771
1
vote
0
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115
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Existence of moment-constrained maximum entropy distribution with support $[0,1]^n$
Given a finite set of moment values $\{\mu_1,\ldots,\mu_N\}$, for which the multi-dimensional finite Hausdorff moment problem is determined. That is, we know that at least one distribution $\mathcal{D}...
- 11
2
votes
0
answers
41
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Distribution of a multivariate continuous process determined by that of linear combination of its coordinates?
To keep the question short: Let $C([0,1], \mathbb{R}^d)$, $d \geq 2$ be the space of all $\mathbb{R}^d$-valued continuous processes. $X$ and $Y$ are two $C([0,1],\mathbb{R}^d)$-valued random variates, ...
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6
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1
answer
3k
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Measurable functions with non measurable image
I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable.
This is motivated by the question Is measure preserving function almost surjective?, ...
- 1,805
1
vote
0
answers
1k
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Sigma algebra of stochastic process
A stochastic process is a collection $(X_t)_{t\in T}$ of random variables from a prob. space $(\Omega,\mathcal{F},P)$ to some measurable space $(E,\mathcal{E})$. Now, in order to understand the whole ...
- 11
4
votes
2
answers
469
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Non-measurability of time integral of non-jointly measurable process
I'm teaching a seminar on probability theory and I want to motivate why joint measurability of a stochastic process is important. The following seems to be the canonical counterexample for a process ...
- 1,675
2
votes
1
answer
250
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Absolute continuity of infinite product of probability measures
Let $(A_i,\mathcal{B}_i,\mu_i)$ for $i=1,2,\ldots$ be a sequence of probability spaces. Let $\nu_i$ be another sequence of probability measures on the same underlying measurable spaces. Assume that $\...
- 404
3
votes
2
answers
278
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The disintegration of the convolution of two probability measures
Let $G$ be a topological group with all the topological conditions in order that some form of the disintegration theorem be applicable (for instance, take $G$ metrizable). Let $N$ be normal and closed,...
- 5,407
17
votes
2
answers
1k
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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
2
votes
1
answer
203
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Non-uniqueness in Krylov-Bogoliubov theorem
So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$.
Of course, if $X$ is just a ...
- 24.8k
1
vote
0
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67
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Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$
Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
- 19
2
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1
answer
2k
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Explicitly representing a random variable in terms of indicator functions
Motivation:
I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula.
I want to prove the change of variable formula (you ...
- 247
1
vote
1
answer
75
views
Measurability of kernel on generating set
Suppose $(X, \Sigma_X)$ and $(Y, \Sigma_Y)$ are measurable spaces such that $\Sigma_Y$ is generated by a set $B$.
Suppose $k : X \times \Sigma_Y \to [0, 1]$ has the property that $k(x, -)$ is a (...
- 239
0
votes
1
answer
121
views
Approximation of a measure on $\mathbb{R}^d$
Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix:
$$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$
I'm trying to prove the existence of a probability measure ...
user122137
7
votes
2
answers
1k
views
Conditional Expectation for $\sigma$-finite measures
Someone knows of some definition or reference of how to define conditional expectation for a measure space with $\sigma$-finite measure.
I think it should be as follows:
Let $(X,\mathcal{B},\nu)$ ...
- 193
2
votes
1
answer
1k
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Understanding measure-preserving transformation [closed]
Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why ...
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4
votes
1
answer
302
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Zero-one law for an independence-like structure
I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false".
Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\...
- 2,207
-1
votes
1
answer
76
views
transformation of two measures on different space
Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.
Let $\Sigma_U$ be the smallest $\sigma-$ field ...
- 1
4
votes
1
answer
188
views
Absolute continuity of measures - reference sought
For two measures $\mu, \nu$ on the same space say that $\mu$ is absolutely continuous with respect to $\nu$ ($\mu \ll \nu$) whenever $\nu(A)=0$ implies that $\mu(A)=0$ too.
Let $(\Omega, \mathsf P$) ...
- 11.3k
2
votes
1
answer
122
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Why do we define the Doléan measure of a continuous square-integrable martingale only on the predictable sets?
If $M$ is a continuous square-integrable martingale on a filtered probability space $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]}\operatorname P)$ and $[M]$ denotes the quadratic variation of $M$, ...
- 167
0
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1
answer
113
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a continuity question concerning metrics on probablility measures
For a metric space $M$, I'll write $Prob(M)$ for the Borel probability measures on $M$.
I am interested in metrics on $Prob(M)$, such as the Kantorovich distance (or other metrics).
If $f: M \...
- 61
2
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1
answer
309
views
Density in Wasserstein space
I am wondering whether the following result is true:
Let $\mathcal W_p(\mathbb R^d)$ be the Wasserstein space of order $p$ and let $\eta$ and $\gamma$ be two probability measures in $\mathcal W_p(\...
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4
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0
answers
2k
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Does rate of convergence in probability come from a metric?
In general, when we talk about convergence of a sequence, we need a topological space. If we want to talk about a rate of convergence, we need to quantify how far away one element of the sequence is ...
- 63
1
vote
0
answers
94
views
Measure of the boundary of the support of a certain function defined by an expectation
Suppose:
$\mathcal{S} = \{ S \in \mathbb{R}^d \ | \ S_i > 0, \forall i = 1,...,d \} $
$R$ is a random vector (on some probability space, $\Omega$) such that, $R: \Omega \to \mathcal{S}$.
$h : ...
- 111
5
votes
2
answers
709
views
Absolute continuity of measures on infinite binary sequences
Suppose $P$ and $Q$ are two probability measures on the space $\Omega = \{0,1\}^{\mathbb N}$ of infinite binary sequences equipped with the product $\sigma$-algebra generated by its cylinder sets, ...
- 51
4
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1
answer
688
views
Locally finite measures on a Polish space form a Polish space
I am looking for a reference where the following question is answered (hopefully affirmatively):
Let $S$ be a Polish space (maybe one needs to assume local compactness?). Is the space of locally ...
- 725
2
votes
1
answer
161
views
Linking error probability based on total variation
Consider probability measure $\mu_{XY}$ defined on $\mathbb{R}^d \times \{1,2,3\}$, and sub-probability measures $\mu_1$, $\mu_2$, and $\mu_3$ as $\mu_1(A):=P(X\in A, Y=0)$ and $\mu_2(A):=P(X\in A, Y=...
- 482
3
votes
1
answer
1k
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Measurable functions in product space
I am reading a book by Billingsley (convergence of probability measures) and he makes a footnote on page 27 which I am struggling to understand. I'll explain the setup below.
Suppose $(X_n,Y_n)$ are ...
- 33
1
vote
0
answers
96
views
Random projection increases the distance?
Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
- 482
3
votes
1
answer
420
views
measurable selection and values of optimization problem
In general, my problem can be formulated as follows: Let $X$ be a random variable with value in $\mathbb R^2$, and let $G:\mathbb R^2 \times \mathbb R\rightarrow \mathbb R$ be a function which is ...
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8
votes
4
answers
2k
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Is every probability measure a pushforward of Lebesgue measure?
If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$?
($\mu$ is ...
- 83
5
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1
answer
945
views
Has a discrete/quantum theory of probability based on the Cournot-Borel principle or something been developed?
In 1930, Émile Borel, the father of measure theory together with his student Lebesgue and a world-class expert in probability theory, published a short note Sur les probabilités universellement ...
- 1,026
3
votes
1
answer
232
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Is there a canonical uniform probability measure on compact subsets of Banach spaces?
One can construct a finite measure on a compact metric space $(X,d)$ by the following procedure:
Fix a non-negative sequence $\{\epsilon_n\}$, $\epsilon_n \to 0$. Let $Y_{\epsilon_n}$ be the minimal ...
- 31
8
votes
2
answers
374
views
Extending the product measure on $2^\omega$
Consider the standard completed product measure $P$ on $\Omega=\{0,1\}^\omega$ corresponding to an i.i.d. sequence of fair coin-flips.
Given $n\in\omega$, let $\rho_n$ be the bijection of $\Omega$ ...
- 2,555
3
votes
1
answer
214
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Inverting the cumulative probability function to find roots of stochastic function
Given a function:
$$f[x]=a\, \Phi \left[-x+\sigma \sqrt{\tau}\right]-\left(b+c\, e^{-d \tau}\right)\Phi \left[-x\right]$$
where $\Phi$ is the cumulative density function of the standard normal ...
7
votes
0
answers
549
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Counter-example to the completeness of the Wasserstein metric
$\newcommand{\P}{\mathcal{P}}$
Let $(E,d)$ be a complete metric space, let $\P(E)$ be the set of all probability measures on $(E,\mathcal{B}(E))$. Let $W_d$ be the $1$-Wasserstein (Kantorovich) ...
- 931
6
votes
2
answers
701
views
Wiener Measure measure on functions?
I know that the Wiener measure for the Brownian motion $\{B_t\}_{t\ge 0}$ on the probability space $(\Omega, \mathscr{F},P)$ can be defined as $\mu=P\circ B^{-1}$ acting on the sigma-algebra generated ...
20
votes
1
answer
2k
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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
1
vote
1
answer
649
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Extreme Points of a set of distributions with moment and/or support constraint
Let $X$ be a random variable with the distribution $F$ (cdf).
What are the extreme points of the sets of the form:
\begin{align}
P_1&=\left\{ F: \int |x|^k dF\le c \right\},\\
P_2&=\left\{ F:...
- 671
3
votes
3
answers
656
views
Free probability with unbounded random variables?
This is partially inspired by this question and this blog post.
When trying to express classical probability in the "free probability" setting one takes an algebra of random variables equipped with ...
- 677
1
vote
1
answer
201
views
Measure of bounded fourth (and below) moment distributions?
Many results in probability theory/random matrix theory/etc require probability distributions with finite fourth moments; what is the measure of such probability distributions (in the space of ...
- 118
4
votes
1
answer
352
views
Measure of the rate of convergence for filtration and conditional expectations
This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that ...
- 869
2
votes
0
answers
130
views
A question on probability measure on the unit ball of Banach spaces
Let $X$ be a Banach space and let $(x^{*}_{n})_{n}$ be a sequence in $X^{*}$. Suppose that $\sum_{n}|\langle x^{*}_{n},x\rangle |\leq \|x\|$ for all $x\in X$.
Question: Is there a probability measure ...
- 3,343
3
votes
1
answer
156
views
Measurability of a particular set generated by discrete probability measures
Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably ...
- 33
8
votes
3
answers
934
views
Question about Wasserstein metric
Let $\mu$ and $\nu$ be two probability measures on $\mathbb R^n$ with finite first moment. Denote by $d:=W_1(\mu,\nu)$, where $W_1(\cdot,\cdot)$ stands for the Wasserstein distance of order $1$.
My ...
- 345
3
votes
1
answer
732
views
What does $\pi$ in the term $\pi$-system stand for?
In measure theory, what does the $\pi$ in $\pi$-system stand for? Also, what about the $\lambda$ in $\lambda$-system? I want to know why Dynkin chosen these names, and why these names make sense.
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