All Questions
Tagged with pr.probability measure-theory
823 questions
21
votes
3
answers
6k
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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 ...
- 8,992
2
votes
1
answer
263
views
Schwartz space on $\bigcup_{n=1}^CR^n$
I have an application where I need to work with the following idea.
Let the space $\bigcup_{n=1}^C \mathbb{R}^n$ be associated with the metric $d$ such that for $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,...
- 21
4
votes
0
answers
756
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Tangent space and gradient on subspace of Wasserstein space given by finitely supported measures
Let $\mathcal{P}_2(M)$ be the 2-Wasserstein space over some Riemannian manifold $(M,g)$ (connected, complete, and without boundary). Let $\mathcal{FP}_2(M,n)$ be the subspace of probability measures ...
- 1,675
4
votes
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 ...
CommunityBot
- 1
1
vote
1
answer
215
views
are there measure preserving mapping in this case?
Suppose f and g are two Borel function on [0, 1]. The push-forward of the Lebesgue measure on [0,1] by f and by g are the same. Then are there some Borel measurable function from [0,1] to [0,1], ...
- 41.1k
0
votes
1
answer
310
views
Questions on a new definition of continuous multivariate distribution
For a univariate distribution or a univariate random variable, we call it continuous/absolutely continuous if its cumulative distribution function (CDF) is continuous/absolutely continuous. Now I am ...
- 17.2k
2
votes
1
answer
133
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Optimal-score partitions
The question about throwing darts asked on the MathOverflow page Sacred Geometry of Chance was not well received, apparently because of "[t]oo much noise around the actual math", as stated in a well-...
- 128k
2
votes
0
answers
117
views
Probability bound involving random, convex sets
Let $X,Y$ and $Z$ be random vectors in $\mathbb{R}^d$ defined on $(\Omega,\mathcal{H},\mathsf{P})$ s.t. $Z$ is $\mathcal{F}$-measurable for some $\mathcal{F}\subset\mathcal{H}$. Define a family of ...
- 185
3
votes
2
answers
278
views
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,...
CommunityBot
- 1
3
votes
2
answers
562
views
Unusual augmentation of a filtration
consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...
- 5,309
3
votes
2
answers
359
views
Non-probabilist term for conditional expectation?
When writing an article I encounter what is essentially a conditional expectation - function defined on a bounded interval (not necessarily of unit length) with Lebesgue measure, but information about ...
- 12.7k
3
votes
0
answers
243
views
Parametric distances on product spaces of measures
Disclaimer: Please excuse my loose language. I'm neither an expert in geometry nor probability. Please ask for clarification if something appears unclear or awkward to you.
Let $X$ be a topological ...
- 6,853
4
votes
2
answers
469
views
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 ...
CommunityBot
- 1
1
vote
1
answer
632
views
Does sequence almost sure convergence imply almost sure convergence?
This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here.
Suppose $x(t,\omega): [0,T]\times\Omega\...
- 23.2k
3
votes
0
answers
126
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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 ...
CommunityBot
- 1
6
votes
1
answer
661
views
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'} ...
- 9,641
6
votes
2
answers
502
views
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 ...
- 11.6k
2
votes
1
answer
2k
views
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
0
answers
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
views
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, ...
- 223
6
votes
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
views
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
2
votes
1
answer
250
views
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 $\...
- 17k
2
votes
1
answer
203
views
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 ...
- 17k
1
vote
0
answers
67
views
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
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 (...
- 128k
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 ...
- 128k
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 $(X,\mathcal{B},\nu)$ ...
- 37.8k
4
votes
1
answer
302
views
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 $(\...
- 29.8k
2
votes
1
answer
1k
views
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 ...
- 128k
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 ...
- 128k
-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 ...
- 3,085
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 $(\Omega, \mathsf P$) ...
- 128k
2
votes
1
answer
122
views
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 $M$, ...
- 138
0
votes
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 \...
- 25.3k
2
votes
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 $\mathcal W_p(\...
- 121
4
votes
0
answers
867
views
For what sub-$\sigma$-algebra are these two measures equivalent?
In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are ...
- 1,228
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 ...
4
votes
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
10
votes
1
answer
3k
views
Applications of Banach-Tarski Paradox to Probability Theory?
I was just curious, since the B-T paradox is a measure theoretic result, if there are any consequences of this paradox in probability theory? Also, is there is a way of stating the B-T paradox in the ...
- 3,574
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, ...
- 61.9k
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 ...
- 11.3k
6
votes
3
answers
938
views
Uniformly distributed sequence in $\mathbb{R}$
We say that a sequence $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ is "uniformly distributed in $[a,b]$", with $a < b$, if $(x_n)_{n=1}^\infty \cap [a,b] \neq \varnothing$ and
$$\lim_{N \to \infty} \...
- 4,713
1
vote
0
answers
96
views
Random projection increases the distance?
Consider two absolutely continuous random variables $X: \Omega \mapsto \mathbb{R}^d$ and $Y: \Omega \mapsto \mathbb{R}^d$ for probability spaces $(\Omega, \mathcal{F},p_X)$ and $(\Omega, \mathcal{F},...
- 482
3
votes
1
answer
420
views
measurable selection and values of optimization problem
In general, my problem can be formulated as follows: Let $X$ be a random variable with value in $\mathbb R^2$, and let $G:\mathbb R^2 \times \mathbb R\rightarrow \mathbb R$ be a function which is ...
8
votes
4
answers
2k
views
Is every probability measure a pushforward of Lebesgue measure?
If $m$ is a probability measure on a measurable space $(X, \Sigma)$, is there necessarily a measurable function $f : [0, 1] \to X$ such that $m(A) = \mu(f^{-1}(A))$ for all $A \in \Sigma$?
($\mu$ is ...
- 4,713
3
votes
1
answer
232
views
Is there a canonical uniform probability measure on compact subsets of Banach spaces?
One can construct a finite measure on a compact metric space $(X,d)$ by the following procedure:
Fix a non-negative sequence $\{\epsilon_n\}$, $\epsilon_n \to 0$. Let $Y_{\epsilon_n}$ be the minimal ...
- 71
3
votes
1
answer
214
views
Inverting the cumulative probability function to find roots of stochastic function
Given a function:
$$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 ...
- 188k