All Questions
Tagged with gn.general-topology pr.probability
79
questions
17
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 $(X, d_X)$ ...
- 2,209
17
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 ...
- 8,322
16
votes
2
answers
3k
views
Is the space of continuous functions from a compact metric space into a Polish space Polish?
Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ ...
user6096
13
votes
1
answer
735
views
Idempotent measures on the free binary system?
Let $(S,*)$ be the free (non associative) binary system on one generator (so $S$ is just the set of terms in $*$ and $1$). There is an extension of $*$ to the space $P(S)$ of finitely additive ...
- 3,607
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: ...
- 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 ...
- 8,322
8
votes
1
answer
351
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 $\mathcal{P}(X, \tau)$?
user76098
8
votes
1
answer
1k
views
Topological necessary and sufficient condition for tightness
Recall the definition of tightness for a probability measure $\mathbb P$ on the Borel $\sigma$-algebra of a metric space $(S,d)$:
For each $\varepsilon>0$, we can find a compact subset $K$ of $X$...
- 3,863
7
votes
1
answer
230
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 ...
- 243
7
votes
2
answers
402
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 ...
- 8,322
7
votes
1
answer
841
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 ...
- 680
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 ...
- 899
7
votes
0
answers
277
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,322
6
votes
1
answer
326
views
Trasportation metric (AKA Earth-Mover's, Wasserstein, etc.) as "natural" / "induced"?
Context: Given a discrete finite metric space $X$ (in my case X={0,1}$^n$ with the Hamming/L$_1$ distance), I need to define the natural or canonical metric on the set of all probability distributions ...
- 101
6
votes
1
answer
382
views
Does a metric refine the weak-* topology on a dual space?
Let $X$ be a topological affine space over $\mathbb C$, with no additional assumptions. Let $X^*$ denote its dual space of continuous affine functionals $X \to \mathbb C$, equipped with the weak-$*$ ...
- 8,322
6
votes
0
answers
183
views
Pettis Integrability and Laws of Large Numbers
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...
- 8,322
6
votes
0
answers
679
views
What is the structure of a space of $\sigma$-algebras?
Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
- 8,322
5
votes
2
answers
490
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 ...
- 8,322
5
votes
1
answer
202
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 ...
- 658
5
votes
1
answer
362
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 ...
- 8,322
4
votes
2
answers
707
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 ...
- 512
4
votes
1
answer
221
views
Statistical models in terms of families of random variables
A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$.
Suppose that $\Theta$ and $...
- 8,322
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\...
- 1,595
4
votes
1
answer
503
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 $...
- 8,322
4
votes
0
answers
94
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 \...
- 12.3k
4
votes
1
answer
525
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
$$\int f d \mu_n \to \int f d\mu$$
...
- 253
3
votes
1
answer
856
views
Convergence in probability only depends on topology?
Suppose $(S,d)$ is a Polish space, and $X$, $(X_n)$ are random variables such that $X_n \to X$ in probability in $(S,d)$. Now suppose $d'$ is another metric on $S$, giving the same topology. Does $...
- 2,765
3
votes
4
answers
507
views
Better terminology than "equivalence class of functions"
Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for ...
- 8,322
3
votes
1
answer
214
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 ...
- 31
3
votes
1
answer
938
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 ...
- 8,322
3
votes
1
answer
142
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,\...
- 5,001
3
votes
1
answer
75
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 ...
- 5,001
3
votes
1
answer
281
views
Relatively compact sets in Ky Fan metric space
Let $(\Omega,P,\mathcal{F})$ be a probability space. $X$, $Y$ are two random variables. The Ky Fan metric defined as: $d_F(X,Y)=\inf\{\epsilon: P(|X-Y|> \epsilon)<\epsilon\}$ (or $d'_F(X,Y)=E \...
- 485
3
votes
1
answer
269
views
Is it possible for a random nowhere dense closed set to have a positive probability of hitting any given point?
Given a compact metrisable topological space $X$, we write $\mathcal{N}(X)$ for the set of non-empty closed nowhere dense subsets of $X$, which is a Polish space under the topology induced by the ...
- 658
3
votes
0
answers
117
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. ...
- 31
3
votes
1
answer
147
views
Existence of disintegrations for improper priors on locally-compact groups
In wide generality, the disintegration theorem says that Radon probability measures admit disintegrations. I'm trying to understand the case when we weaken this to infinite measures, specifically ...
- 8,322
3
votes
0
answers
143
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 ...
- 170
3
votes
1
answer
971
views
"Relative compactness of a family of probability measures" and relative compactness & sequential compactness of sets
I'm studying Billingsley's convergence of probability measures, and wondering why the definition of "Relative compactness of a family of probability measures" reasonable.
In the discussion ...
- 31
2
votes
2
answers
929
views
measurability of integrated functions
Hello everybody,
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 ...
- 1,505
2
votes
1
answer
259
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 ...
- 395
2
votes
1
answer
187
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 ...
- 24.5k
2
votes
1
answer
135
views
Probability measures on a dense subset
Let $D\subseteq X$ be a dense subset of a separable metric space $X$. Let $P(D)$ and $P(X)$ respectively denote the probability measures on $D$ and on $X$ with their weak topologies. Then, if we ...
- 5,001
2
votes
2
answers
199
views
non-homogeneous counting process
Consider a counting process $\{N(t), t\geq 0\}$ where the time distribution between any two consecutive events, say $k$ and $k+1$ has a Poisson rate $\lambda(k)$, which is an explicit function of $k$....
- 31
2
votes
1
answer
148
views
Polish spaces and isomorphisms
An isomorphism between two measurable spaces $(X_1,\mathcal{B}_1), (X_2,\mathcal{B}_2)$ is a measurable bijection $f:X_1\rightarrow X_2$ whose inverse is also measurable.
QUESTION. Can there be an ...
- 41.2k
2
votes
1
answer
533
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 ...
- 8,322
2
votes
1
answer
324
views
General topology book recommendation for advanced probability theory
I would like to know if anyone could suggest a general topology book for a deeper understanding of probability at advanced level. If there is an advanced topology book oriented to probabilists, I ...
2
votes
3
answers
396
views
Looking for a reference: $f$-divergences are lower semicontinuous
I know that the weak lower semi-continuity of the KL divergence was proved in [1]. If I remember well, the same property is true for any $f$ divergence (with suitable assumptions on the probability ...
- 271
2
votes
1
answer
269
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}...
- 271
2
votes
1
answer
127
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 $\mathcal{P}_{n:+}(\...
- 5,001
2
votes
1
answer
374
views
When is a space of probability measures not perfectly normal?
I am looking for examples of pairs ($(\Omega,\Sigma)$, ($\mathcal P(\Omega)$, $\tau$)), where $(\Omega,\Sigma)$ is a measurable space and ($\mathcal P(\Omega)$, $\tau$) is a space of probability ...
- 23