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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 ...
Tom LaGatta's user avatar
  • 8,512
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 $(X, d_X)$ ...
Nate Ackerman's user avatar
9 votes
1 answer
4k views

What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here. The Wikipedia article on convergence of measures defines three kinds of convergence: ...
user39080's user avatar
  • 203
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 ...
Tom LaGatta's user avatar
  • 8,512
8 votes
1 answer
360 views

Can we recover a topological space from the collection of Borel probability measures living on it?

Let $(X, \tau)$ be a topological space, and $\mathcal{P}(X, \tau)$ be the Borel probability measures living on $X$. Can we recover $(X, \tau)$ from $\mathcal{P}(X, \tau)$?
user avatar
7 votes
1 answer
261 views

Comparison of several topologies for probability measures

Let $X$ be a compact metric space and denote $\mathcal M(X)$ the set of probability measures on $X$. For $\mu\in\mathcal M(X)$ we write $\operatorname{supp} \mu$ for the support of $\mu$. As is well ...
Kass's user avatar
  • 243
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 ...
Tom LaGatta's user avatar
  • 8,512
7 votes
1 answer
1k views

Reference request: norm topology vs. probabilist's weak topology on measures

Let $(X,d)$ be a metric space and $\mathcal{M}(X)$ be the space of regular (e.g. Radon) measures on $X$. There are two standard topologies on $\mathcal{M}(X)$: The (probabilist's) weak topology and ...
JohnA's user avatar
  • 710
7 votes
0 answers
3k views

What is vague convergence and what does it accomplish?

For convenience, let's say that I have a locally compact Hausdorff space $X$ and am concerned with probability measures on its Borel $\sigma$-algebra $\mathcal{B}(X)$. Natural vector spaces to ...
Greg Zitelli's user avatar
  • 1,084
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 ...
Tom LaGatta's user avatar
  • 8,512
6 votes
0 answers
196 views

Prokhorov's theorem for countably many random measures on a Polish space

I am looking for help to show the following lemma: Lemma Let $(\Omega,\mathcal A,\mathbb P)$ be a complete, standard Borel probability space and $\mathcal X$ a Polish space. Let $\mathcal P(\mathcal ...
Pierre-François Massiani's user avatar
5 votes
1 answer
254 views

Is the topology of weak+Hausdorff convergence Polish?

Let $X$ be a compact metric space, $P_X$ the set of Borel probability measures on $X$, and $K_X$ the set of non-empty closed subsets of $X$. I will define the "topology of weak+Hausdorff ...
Julian Newman's user avatar
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 ...
Tom LaGatta's user avatar
  • 8,512
4 votes
2 answers
714 views

Polish by compact is Polish?

Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish? I have a specific space in mind, so if the ...
biringer's user avatar
  • 532
4 votes
1 answer
360 views

Measurability of Markov kernel wrt the Borel $\sigma$-algebra generated by the weak topology

Consider two Polish metric probability spaces $(\mathcal{A}, \Sigma_\mathcal{A})$ and $(\mathcal{B}, \Sigma_\mathcal{B})$, endowed with their Borel $\sigma$-algebras. Denote as $\mathcal{P}_\mathcal{B}...
ECL's user avatar
  • 345
4 votes
1 answer
1k views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
SBF's user avatar
  • 1,655
4 votes
1 answer
579 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 $...
Tom LaGatta's user avatar
  • 8,512
4 votes
0 answers
119 views

Is the range of a probability-valued random variable with the variation topology (almost) separable?

Let $X$ and $Y$ be uncountable Polish spaces, $\Delta(Y)$ be the space of Borel probability measures on $Y$ endowed with the Borel $\sigma$-algebra induced by the variation distance, and let $g:X\to \...
Michael Greinecker's user avatar
4 votes
1 answer
565 views

convergence of integral for each bounded function in probability

Let $\mu, \mu_1, \mu_2, \dots$ be random measures on a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$. Suppose I know that $$\int f d \mu_n \to \int f d\mu$$ ...
Valentas's user avatar
  • 255
3 votes
2 answers
994 views

measurability of integrated functions

DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a homework, as ...
user avatar
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 ...
shasha's user avatar
  • 31
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 ...
Tom LaGatta's user avatar
  • 8,512
3 votes
1 answer
143 views

Density of $C(X,\operatorname{co}\{\delta_y\}_{y \in Y})$ in $C(X,\mathcal{P}(Y))$

Let $X,Y$ be locally-compact Polish spaces, equip the set $\mathcal{P}(Y)$ of probability measures on $Y$ with the weak$^{\star}$ topology (topology of convergence in distribution), and equip $C(X,\...
ABIM's user avatar
  • 5,405
3 votes
1 answer
77 views

Continuous selection parameterizing discrete measures

Let $\mathcal{P}_n(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$. Then any measure in $\mathcal{P}_n(\mathbb{R})$ is in the image of ...
ABIM's user avatar
  • 5,405
3 votes
0 answers
144 views

What is an example of a non-tight probability measure?

Billingsley (Convergence of Probability Measures, 1968) and van der Vaart and Wellner (Weak Convergence and Empirical Processes, 2023) discuss the concept of tight probability measures and use the ...
cgmil's user avatar
  • 277
3 votes
0 answers
145 views

Eigenvalues of random matrices are measurable functions

I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable. If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
Curtis74's user avatar
3 votes
0 answers
174 views

Any reference on Jensen inequality for measurable convex functions on a Hausdorff space?

I asked this question on math.stackexchange and I was suggested that asking it may be more appropriate. This is part of my research which tries to extend some of Choquet's theory to some non-compact ...
P. Quinton's user avatar
2 votes
1 answer
351 views

Measurability of integrals with respect to different measures

Let $Y$ be a locally compact Hausdorff topological space (further assumptions like metrizability, separability, etc., may be added if necessary) and let $\mathscr Y$ denote the Borel $\sigma$-algebra ...
triple_sec's user avatar
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 ...
Bjørn Kjos-Hanssen's user avatar
2 votes
1 answer
556 views

Is this a closed set?

Let $\Theta$ and $X$ be two (Hausdorff) topological spaces. Let $\mathbb P : \Theta \to \Delta(X)$ be a "statistical model", i.e., a continuous function from parameter space $\Theta$ to the space of ...
Tom LaGatta's user avatar
  • 8,512
2 votes
1 answer
155 views

Covering of discrete probability measures

Let $\mathcal{P}_{n:+}(\mathbb{R})$ denote the set of probability measures on $\mathbb{R}$ for the form $\sum_{i=1}^n k_i \delta_{x_i}$ where $k_i>0$. Then any measure in $\mathcal{P}_{n:+}(\...
ABIM's user avatar
  • 5,405
2 votes
1 answer
414 views

When is a space of probability measures not perfectly normal?

I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability ...
user1211719's user avatar
2 votes
1 answer
245 views

Probability measures on $L^p$

Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
Tom LaGatta's user avatar
  • 8,512
2 votes
0 answers
49 views

$\sigma$-compactness of probability measures under a refined topology

Denote Polish spaces $(X, \tau_x)$ and $(Y, \tau_y)$, where $X$ and $Y$ are closed subsets of $\mathbb{R}$. Consider a Borel measurable function $f: (X \times Y, \tau_x \times \tau_y) \rightarrow \...
Hans's user avatar
  • 195
2 votes
0 answers
136 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 $\phi(x):=1+|x|$, $$ \...
CodeGolf's user avatar
  • 1,835
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}\...
user avatar
1 vote
1 answer
331 views

Extension of measurable function from dense subset

Let $M$ be a compact riemannian manifold equipped with a geodesic distance and let $\mathcal{B}(M)$ be the borel sigma algebra generated by the geodesic distance. Let $(\Omega,\mathcal{F},\mathbb{P})$...
Giuseppe Tenaglia's user avatar
1 vote
1 answer
172 views

A question about pushforward measures and Peano spaces

Specifically my question is the following: Let $P$ be a Peano space. If $(P,\sigma,\mu)$ and $(P,\sigma,\nu)$ are both nonatomic probability measures, does there exist a continuous function $f:P\to P$ ...
O-Schmo's user avatar
  • 33
1 vote
0 answers
52 views

A local base for space of probability measures with Prohorov metric

Let $S$ be a Polish space. Let $P(S)$ denote the space of probability measures on $(S,\mathcal{B})$, where $\mathcal B$ is the Borel-$\sigma$-algebra over $S$. Equip $P(S)$ with the Prohorov metric. I ...
Error 404's user avatar
  • 111
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\...
PepitoPerez's user avatar
1 vote
0 answers
260 views

Generating the sigma algebras on the set of probability measures

I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
Mark's user avatar
  • 11
0 votes
1 answer
282 views

Explicit examples of (probability) measures on $\prod \mathbb{R}$

Let $\prod_{n \in \mathbb{N}} \mathbb{R}$ be equipped with the Tikhonov product of the Euclidean topologies on $\mathbb{R}$ and let $B$ the corresponding Borel $\sigma$-algebra. What is are some ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
161 views

question about the tightness of probability measures for a general topological space

Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
CodeGolf's user avatar
  • 1,835
-1 votes
1 answer
148 views

Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]

Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology. Let $X$ be a topological space (for convenience, it might be Polish ...
user66910's user avatar