All Questions
Tagged with measure-theory pr.probability
823 questions
0
votes
0
answers
93
views
Regularity with respect to the Lebesgue measure through dimensions
Let us consider two probability measures $\mu \in \mathcal{P}(\mathbb{R}^{p})$ and $\nu \in \mathcal{P}(\mathbb{R}^{q})$ with $p,q \in \mathbb{N}^{*}$. We note $\#$ the push forward operator i.e for $...
- 407
2
votes
1
answer
294
views
Information theory for uncountably infinite-dimensional continuous random variable
I'm exploring the possibility to apply information theory on an uncountably infinite-dimensional scenario. I found the concept of generalized entropy for continuous random variables defined on finite-...
- 263
0
votes
1
answer
55
views
Looking for a family of random variables such that only the second clause is fulfilled [closed]
Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if
i) $sup_{i \in I} E(X_i) <\infty$
ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
- 11
-1
votes
1
answer
396
views
Convergence of Radon-Nikodým derivative
Imagine we have a sequence of finite measures $\nu_n << \mu_n$ (on the torus $\mathbb{T}^2\subseteq \mathbb{R}^2$) converging weakly to some measures $\nu << \mu$. Do we automatically have ...
- 183
4
votes
2
answers
615
views
Convergence of conditional measures for a convergent sequence of probabilities whose projection is constant
Setting
Suppose $\mu_n$ is a sequence of probability measures on $[0,1]\times [0,1]$ converging to a limit probability $\mu$ meaning that
$$ \lim_{n\to+\infty}\int f(x,y)d\mu_n(x,y) = \int f(x,y)d\mu(...
- 4,304
2
votes
1
answer
159
views
Can we show that this transition semigroup preserves a certain Wasserstein space?
Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
- 167
1
vote
1
answer
189
views
If a Markov semigroup is eventually contractive, can we conclude that it admits a unique invariant measure?
Let $E$ be a separable $\mathbb R$-Banach space, $\rho$ be a complete separable metric on $E$, $\operatorname W_\rho$ denote the Wasserstein metric of order $1$ associated to $\rho$, $\mathcal M_1(E)$ ...
- 167
1
vote
1
answer
183
views
If $(κ_t)$ is a semigroup with invariant measure $\mu$ and $ν$ is singular to $\mu$, then $νκ_t$ might not converge to $\mu$ in total variation norm
Let $E$ be a Polish space, $(\kappa_t)_{t\ge0}$ be a Markov semigroup on $(E,\mathcal B(E))$, $\mu$ be a probability measure on $(E,\mathcal B(E))$ invariant with respect to $(\kappa_t)_{t\ge0}$ and $\...
- 167
0
votes
1
answer
109
views
Can we say that $f_\infty(t)=g_\infty(t) \text{ a.e}$?
Let $(E,\mathcal{A},\mu)$ be a finite measure space. Let $\{f_n\}$ and $\{g_n\}$ two uniformly integrable sequences such that:
$$
\qquad\forall n\geq 1~:~ f_n(t)=g_n(t)~~ \text{ a.e.}\tag1
$$
$$
\...
- 199
-4
votes
1
answer
66
views
Can we say that $\{f_n\}\text{ is uniformly integrable over }E\setminus (\cap_p B_p)$? [closed]
Let $(E,\mathcal{A},\mu)$ be probability space and $\{f_n\}$ be sequence of functions such that
$$
\sup_n\int_{E}|f_n|d\mu<+\infty.
$$
Let $\{B_p\}$ be a sequence non-increasing in $\mathcal{A}$...
- 115
0
votes
1
answer
144
views
Integration against a measure that has an integral form
Suppose that $(X, \mathcal{X})$ is a measurable space and $(Y,\mathcal{Y}, \mu)$ is a measure space (in my particular application, they are Polish spaces endowed with their Borel $\sigma$-algebra). ...
user159835
5
votes
2
answers
2k
views
Tight sequence of measures
This is probably a very easy question for experts in probability or measure theory.
I have a sequence of finite measures $\mu_{n}$ on a non-compact metric space $X$ such that $\mu_{n}$ converges to $\...
- 38
2
votes
1
answer
122
views
How is this bound for a Wasserstein contraction coefficient in this paper obtained?
I'm trying to understand the following conclusion from this paper (see below for the relevant paragraphs):
I'm not sure whether they really mean that it follows from the statements of Lemma 3.2 (...
- 167
5
votes
2
answers
589
views
Properties of measures that are not countably additive but have countably additive null ideals
This is a very naive question, maybe more of a reference request than anything else.
Let $(X, \mathcal X)$ be a measurable space. If $m$ is a real-valued function on $\mathcal X$, we say that $m$ has ...
- 869
0
votes
1
answer
683
views
The properties of total variation metric
The questions was asked by me on Math StackExchange, but no answer appears, so I ask for help again.
Let $(X, d)$ be a complete (Hausdorff, separable, local compact and other nice properties you want)...
- 1,799
1
vote
1
answer
66
views
Can this be translated to a truncated multivariate moment problem?
Fix $\mathbf t \in \mathbb{R}_+^d$. I am looking for a r-atomic measure $\nu$ on $\mathbb{R}_{+}^{d}$ that solves the following 'moment' conditions.
$$\forall\, \mathbf i \in \mathbb{N}^{d} \, s.t \, ...
- 686
1
vote
0
answers
36
views
Does a total preorder on lotteries that preserves countable mixtures preserve arbitrary mixtures?
Let $X$ be a countable set. A lottery on $X$ is a function $\lambda: X \to [0,1]$ such that $\sum_x \lambda(x) = 1$. Let $\Delta X$ be the set of lotteries on $X$.
A total preorder $\preceq$ on $\...
- 869
-3
votes
2
answers
156
views
Getting almost certainty from uncountably many low-probability events
Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X_{\alpha,n}\}_{a \in A, ...
- 5,405
-1
votes
1
answer
138
views
On the concentration of Lipschitz functions near its expectation, where the vector has identical but not independent, components
Consider the random vector $X:=(X_1\dots X_1) \in \mathbb{R}^n, X_1 \sim \mathcal{N}(0,1).$ Notice the identical components, they're identically distributed but not independent.
Now, I was wondering ...
- 1,467
2
votes
1
answer
70
views
$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $
Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space.
The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by:
$$
\...
- 199
3
votes
1
answer
233
views
A question about finitely additive integration
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space ($\mathbb P$ is countably additive). Let $\{p_\omega: \omega \in \Omega\}$ be a family of (countably additive) probability measures on $(\...
- 869
-1
votes
1
answer
88
views
Can we say that: $ \sum_{n\geq 1}{\frac{1}{n}(f_n(\omega)-g_n(\omega))}<\infty\qquad a.e $
Let $(\Omega,\mathcal{A},\mu)$ be a finite mesure space, and $\{f_n\}$ and $\{g_n\}$ two $L^1$-bounded sequences, such that :
$$
\sum_{n\geq 1}{\frac{1}{n}(F_n(f_n)(\omega)-g_n(\omega))}<\infty\...
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
4
votes
2
answers
855
views
Disintegration, conditional probabilities, and conditional expectation
On the Wikipedia page there is a note that conditional probability measures can be described by disintegration. However, I can seem to find a clear exposée of how this construction is related to ...
- 5,405
0
votes
1
answer
103
views
Can the joint law $P \circ (X,Y)^{-1}$ of two random variables $X$ and $Y$ be written as $P \circ (X,\phi(X,U))^{-1}$ for $U$ uniform in $[0,1]$?
I want to know whether there is some general assumpitons we can make on two measurable spaces $E$ and $F$ (e.g. polish, complete, separable,...) such that we can ensure that the following "Theorem" ...
- 309
2
votes
0
answers
520
views
Example of a non-reflexive Banach space and two sequences
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$.
If $X$ is reflexive, ...
- 199
0
votes
1
answer
87
views
difference between: Measurable multifunction integrably bounded and Measurable multifunction integrable
I read the article "Komlós Theorem for Unbounded Random Sets" by G. KRUPA (MSN), but I did not understand the difference between:
Measurable multifunction integrably bounded,
Measurable multifunction ...
- 117
2
votes
3
answers
255
views
Image of probability measures under measurable mappings
Given two probability measures on two probability spaces, ($\mu, X$) and ($\gamma, Y$), what's the sufficient and necessary condition such that there is a measurable mapping $f:X\rightarrow Y$, such ...
- 71
0
votes
1
answer
79
views
Reduce ergodicity to the ergodicity of the coordinate process
Let $(E,\mathcal E,\lambda)$ be a probability space and $\lambda$ be a measurable map on $(E,\mathcal E)$ with $\lambda\circ\tau^{-1}=\lambda$.
I would like to show that $\tau$ is $\lambda$-ergodic ...
- 167
2
votes
2
answers
328
views
Why the Komlós theorem is not valid for any sequence of measurable functions?
I read an article, and they use a certain theorem, called Komlós theorem, which says:
Theorem 1 (Komlós theorem)
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $ (f_n)_{n\geq 1} \subset ...
- 199
0
votes
1
answer
1k
views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
- 710
1
vote
1
answer
88
views
Convergence of probability measures which (asymptotically) concentrate along a submanifold
Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...
- 801
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,\...
- 5,405
0
votes
1
answer
378
views
Concentration of norm of linearly transformed normal random vector as dimension go to infinity
Earlier asked on MSE, but didn't get an answer, so posting here:
Let $X=(X_1 \dots X_n) \in \mathbb{R}^n, X_i\sim N(0,1), iid.$ Let $B: \mathbb{R}^n \to \mathbb{R}^n $ be the diagonal linear map: $...
- 1,467
6
votes
1
answer
291
views
Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group
Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables :
Let $X$ and $Y$ be $G$-valued ...
- 2,101
1
vote
1
answer
84
views
$ \|u_k-v_k\|_2\leq \min \bigg(\inf_{n\geq k}{\|f_n1_{\{|f_n|\leq k\}}\|_2},\frac{\epsilon_{k-1}}{4k}\bigg) $
Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}_n$ be a sequence of simple functions such that:
\begin{align*}
f_n1_{\{\lvert f_n\rvert\leq k\}}\overset{\sigma(L^2,L^2)}{\underset{n}{\...
- 13
1
vote
0
answers
184
views
Bounding the total variation metric between Gaussian mixtures
Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
- 5,405
3
votes
1
answer
233
views
A subadditive maximal ergodic theorem
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
- 167
0
votes
1
answer
58
views
Good upper-bound for $\mathbb E_A[e^{-t\|A\|_2}]$, for $t\ge0$ and random m by n matrix with iid entries with law $N(0,1)$
Let $A$ be a random $m$-by-$n$ matrix with iid $N(0,1)$ entries, $m$ and $n$ large with $n/m \longrightarrow \alpha \in (0, 1)$ . Let $\|A\|_2$ be the largest singular value of $A$ (i.e the spectral ...
- 6,853
5
votes
1
answer
363
views
Inverse marginal property of a collection of $\sigma$-algebras
In my paper "On the inverse best approximation property of systems of subspaces of a Hilbert space"
I introduced the Inverse marginal property (IMP) for a collection of $\sigma$-algebras.
Let $(\...
- 935
0
votes
1
answer
114
views
Can we say that : $ \exists f_{\infty}\in L_{\mathbb{R}}^{1} \text{ such that: } f_n\to f_\infty\text{ a.e and in } L_{\mathbb{R}}^{1} $ [closed]
Let $(E,\mathcal{A},\mu)$ be a finite measure space and $\{f_n\}\subset L_{\mathbb{R}}^{1}$ such that:
$$
\sum_{i=2}^{\infty}{\int_{E}{|f_n(t)-f_{n-1}(t)|d\mu(t)}}<+\infty
$$
Can we say that :
$$
\...
- 115
0
votes
1
answer
121
views
$\sum_{n=1}^{\infty}{\frac{1}{n^{1+\epsilon}}\mathbb{E}\big((|X_n|\mathbb{1}_{|X_n|\leq n})^{1+\epsilon}\big)}<\infty,~~\forall\epsilon>0 $
Let $(E,\mathcal{A},\mathbb{P})$ be a probability space $\{X_n\}$ be a sequence of random variable, such that:
$$
(1)~.~~~\sup_n\mathbb E (|X_n|)<\infty\Rightarrow
$$
$$
(2)~.~~~\dfrac{M_j}{2}<...
- 115
1
vote
1
answer
176
views
The existence of a copy of a random variable with conditional expectation constraint
Let there be two random variables 𝑋 and 𝑌 with a certain joint copula. Is it always true that there is another random variable 𝑍 independent from 𝑌 such as the vectors $(X,Y)$ and $(X,Z)$ have the ...
- 375
0
votes
1
answer
153
views
$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable
Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space.
Suppose $\{X_n\}$ is a sequence of random variables satisfying :
$$
\sup_{n}{\mathbb{E}(|X_n|)} <\infty
$$
Suppose that
$$
\dfrac{M_j}{...
- 57
1
vote
0
answers
72
views
Local time as a measurable map from Wiener space
Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...
- 891
2
votes
0
answers
35
views
When does a measure-valued map admit a continuously parametrized density function?
Let $X$ and $Y$ be Polish spaces, let $\mathcal{P}(Y)$ be the space of Borel probability measures on $Y$ endowed with the smallest $\sigma$-algebra such that all functions of the form $\nu\mapsto\nu(A)...
1
vote
1
answer
132
views
Lambda system generated by a non-atomic collection
Consider a probability space $(X,\Sigma,P)$.
Let say that a collection $\mathcal{B}\subseteq\Sigma$ is non-atomic if for every $E\in\mathcal{B}$ and $\alpha\in(0,P(E))$ there exists $F\in\mathcal{B}$ ...
- 11
3
votes
1
answer
166
views
Is the inner/outer measure mapping continuous?
Let $\mathcal F$ be a field of subsets of a set $\Omega$. Equip the space $[0,1]^\mathcal F$ of functions from $\mathcal F$ into $[0,1]$ with the product topology. Then, the set $\Delta$ of finitely ...
- 869
2
votes
0
answers
61
views
Measurable extensions of probability measures
Let $X$ be a set, and let $\mathcal G \subset \mathcal F$ be $\sigma$-fields over $X$. Let $\Delta_\mathcal G$ (resp. $\Delta_\mathcal F$) be the set of probability measures on $\mathcal G$ (resp. $\...
- 869
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