Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
1,981
questions
2
votes
0answers
41 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 $(\...
5
votes
1answer
58 views
completeness of $\mathcal M(\Omega)$ without any topological assumptions?
Let $(\Omega,\Sigma)$ be a measurable space (no reference measure is chosen!), and $V$ a finite-dimensional normed vector space.
Note carefully that I am not choosing any topology on $\Omega$, so the $...
1
vote
1answer
89 views
Conditional expectation values defined by expectation values
I asked this question a couple of days ago on Math.SE but without any echo (no upvotes, although I offered a bounty). But because I did it for oversight from a reputable/professional source I now ...
-1
votes
1answer
78 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\...
2
votes
0answers
47 views
Lebesgue measure of set of equidistant points with respect to a finite set
Let $X$ be a finite subset of $\mathbb{R}^n$; equip $d$ with a metric, and let $\emptyset \subset X\subseteq \mathbb{R}^n$ be of cardinality $N>0$.
What requirements on my metric do I need so ...
11
votes
1answer
292 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^...
-2
votes
0answers
30 views
v, an extension of a premeasure will always be so v(E) <=u(E) [closed]
enter image description here
I don't understand the part of the proof that shows that v(E) is always less than or equal to u(E). It appears to me that they are arguing backwards but I don't know how ...
4
votes
0answers
101 views
Continuous disintegration
Given a suitable Borel measure $\mu$ on a suitable topological space $X$ and a Borel function $\pi:X \to Y$, where $Y$ is another suitable topological space, the disintegration theorem gives a Borel ...
-3
votes
0answers
28 views
Defining borel measure on unit circle [closed]
I am going through the section on trigonometric series, chapter 4 of RCA Rudin, where he defines $L^p$ norm of functions defined on the unit circle. However a development of measure on the unit circle ...
3
votes
2answers
213 views
Existence of measurable “inclusion” into Euclidean space
Let $(\Omega,\mathfrak{F})$ be a measurable space. When does there exist an injective measurable function $f:(\Omega,\mathfrak{F})\to (\mathbb{R}^n,B(\mathbb{R}^n))$ to some Euclidean space, here $B(\...
1
vote
2answers
102 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 ...
0
votes
1answer
24 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" ...
2
votes
0answers
81 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, ...
-3
votes
1answer
179 views
Measure Theories with a different convention to $\infty\cdot 0 =0$ [closed]
As we all know in a first course in measure theory we define a symbol $\infty$ to satisfy $\infty \cdot 0=0$, but there are more two possible choices for a convention as someone has shown to me; one ...
-1
votes
1answer
74 views
Isometric stratification preserves volume?
Let $K\subset \mathbb{R}^k$ be a non-empty compact subset let $f:K \to K$ be Lipschitz and surjective. If, moreover, $f$ is an isometry then clearly $f$ preserves the Lebesgue measure of $K$.
I ...
1
vote
2answers
81 views
Locally compact Polish groups acting on standard Lebesgue spaces
If $G$ is a countable discrete group, then one can consider the Bernoulli shift $2^G$. $G$ acts on $2^G$ via shift, and letting $\mu$ be the product of the $(1/2, 1/2)$-measure in each coordinate, ...
1
vote
0answers
66 views
What is the closure of this set in $H^1(\mathbb{R}^2)$?
I'm not sure that if this is a difficult question or not. I asked it on MSE and it hasn't been answered and so I thought I might ask it here:
What is (or how can we describe) the closure in $H^1(\...
2
votes
3answers
138 views
Covering families of sets by small-measure partitions
Let $(X,\mathscr{A},\mu)$ be a probability space and let $\{A_1,\ldots,\}\subset\mathscr{A}$ be a countable family of sets with small measure: say $\mu(A_i)\le\epsilon$. I am trying to show that one ...
2
votes
1answer
53 views
Compact embedding of space of signed Radon measures into Sobolev space $W^{-1,q}$ from Evans paper; Does it work in one space dimension?
Background: I work on a PDE problem where I have some approximating sequence of measure-valued functions and I need to compactly embed it into some negative Sobolev space $W^{-m,q}$ on the bounded ...
0
votes
1answer
49 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 ...
2
votes
3answers
166 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 ...
0
votes
1answer
51 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 ...
3
votes
0answers
141 views
Sets of finite perimeter: intersection with an half space
I have a question regarding sets of finite perimeter. In particular I'm interested to find
$$\mu_{E \cap H_t}, \label{1}\tag{1}$$
where $E$ is a set of finite perimeter in a generic open set $\Omega \...
2
votes
1answer
126 views
Atomless, c-additive measures in ZFC
This is a follow-up question to this one.
Is there a ZFC example of an atomless measure that is $2^\omega$-additive, meaning, fewer than continuum many null sets have measurable union that is null?
-6
votes
0answers
258 views
Analyzing my definition of Average which uses a variation of the Lebesgue Integral and Measure [closed]
Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. As noted in previous questions, I want to define an average using a new measure and integral since I found certain aspects of the ...
3
votes
1answer
150 views
Finitely additive, $\kappa$-additive atomless measures in ZFC
Under Martin's Axiom (and non-CH) the Lebesgue measure is $2^\omega$-additive in the sense that unions of fewer than continuum ($2^\omega$) many null sets are measureable and null. In ZFC we may ...
1
vote
0answers
58 views
“Constructive” proof that compact sets $K\subseteq L_1(\mu\times\nu)$ are contained in products $S\widehat{\otimes} T$
A.Defant and K.Floret in chapter 7 of their Tensor Norms and Operator Ideals prove the equality
$$
L_1(\mu\times\nu)\cong L_1(\mu)\widehat{\otimes}L_1(\nu)
$$
for measures $\mu$ and $\nu$. At the same ...
6
votes
1answer
145 views
Prokhorov theorem on non Polish spaces
It is well known that if $X$ is a Polish space and $\mathcal{F} \subset \mathcal{M}_+(X)$ (the set of finite positive Radon measures on $X$) is uniformly tight and bounded in mass, it is relatively ...
2
votes
2answers
200 views
Why the Komlós theorem is not valid for any sequence of measurable functions?
I read an article, and they use a certain theorem, called Komlós theorem, which says:
Theorem 1 (Komlós theorem)
Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $ (f_n)_{n\geq 1} \...
-2
votes
1answer
173 views
Bounding $L^p$ norms in terms of lower-order $L^q$ norms
Suppose $f,g\in L^q(\Omega)$ ($\Omega\subset \mathbb{R}^n$) for all $1\le q\le p$. Here, $L^p(\Omega)$ is defined with respect to some measure $\mu$ that is absolutely continuous wrt Lebesgue measure. ...
1
vote
1answer
67 views
Convergence of probability measures which (asymptotically) concentrate along a submanifold
Let $V : (-1, 1)^d \to \mathbf{R}_+$ be a smooth function, and for $\beta > 0$, define
\begin{align}
P_\beta ( dx ) &= \exp \left( - \beta V ( x ) \right) / z (\beta) \, dx\\
z (\beta) &= \...
-5
votes
2answers
417 views
Coming up with a equivalent (or close) definition for an average which is easier to compute? [closed]
Continuing from my last question, I understand that my definition is unclear so I have modified it.
Since no one has answered my question on math stack exchange, I decided to ask here.
Definition
...
-1
votes
0answers
43 views
Why there exists sub-sequence of $\{X_n1_{|X_n|\leq 1}\}$ weakly convergent
Let $(E,\mathcal{A},\mu)$ be a probability space and $\{X_n\}$ be a sequence of random variables.
Put $Y_n=X_n1_{|X_n|\leq 1}$.
Why there exists sub-sequence $\{X_{n_i}\}$ of $\{X_n\}$ such that:
$$...
2
votes
0answers
47 views
About the current of finite mass
In Demailly's e-book Complex analytic and differential geometry,
chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ ...
1
vote
1answer
119 views
Integrable functions as elements of closed absolutely convex hulls of precompact sets of indicator functions
I am not a specialist in measure theory, so excuse me if this is simple.
Let $\mu$ be a finite measure on a set $X$ (for example, the Lebesgue measure on $[0,1]$). Integrable functions on $X$ can be ...
1
vote
1answer
66 views
Ergodic decomposition - how does restricting measure effect it? (Choquet Theory)
Suppose that $G$ is a discrete countable group and $\mu$ is an IRS (invariant random subgroup) of $G$: $\mu$ is conjugation invariant as a probability measure on the subgroups of $G$.
Since all the $...
7
votes
1answer
108 views
When is the Radon-Nikodym derivative locally essentially bounded
Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I_K)<\infty$ for every ...
6
votes
1answer
80 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,\...
0
votes
1answer
49 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: $...
5
votes
1answer
105 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 ...
1
vote
1answer
46 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:
$$
f_n1_{\{|f_n|\leq k\}}\overset{\sigma(L^2,L^2)}{\underset{n}{\longrightarrow}} u_k\...
1
vote
0answers
38 views
Bounding the total variation metric between Gaussian mixtures
Let $\mathcal{P}(\mathbb{R}^d)$ the space of probability measures on $\mathbb{R}^d$ with total variation metric $\delta$, fix $k \in \mathbb{N}$, and let $\mathcal{P}'\subset \mathcal{P}(\mathbb{R}^d)$...
2
votes
1answer
138 views
A subadditive maximal ergodic theorem
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
5
votes
1answer
163 views
How much time does a function spend above or below its average value around a point?
Given a locally integrable function $f: \mathbb R \to \mathbb R$, define $
K: \mathbb R \times \mathbb R+ \to \mathbb R$ by
$$
K(x, r) :=
\begin{cases}
1, & \text{if }f(x) > \dfrac{1}{2r}\...
8
votes
2answers
224 views
Graph metric approximating Euclidean metric
I've been reading Wolfram's recent articles about graph/mesh/grid structures as an analogy for physical space, and it seems to me that there will be a problem getting the notion of distance to work ...
18
votes
1answer
435 views
Acting with a finite number of rotations on a set of positive measure can you fill almost the whole circle?
Let $E\subset S^1$ have positive Lebesgue measure. Do there exist finitely many rotations
$r_1, r_2, \dots ,r_n$ such that $r_1E\cup r_2E\cup \dots\cup r_nE$ has measure $2\pi$? Or is there a ...
0
votes
1answer
34 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
votes
3answers
438 views
Acting with all rational rotations on a subset of the circle having positive measure do you fill almost the whole circle?
Set $\Gamma$ for the group of the roots of the identity: $\Gamma=\{z\in \Bbb C | z^n=1$, for some $n\geq 0\}$ and for $E\subset S^1$ set
$\Gamma E=\{z\zeta, z\in \Gamma, \zeta\in E \}$
A trivial but ...
4
votes
1answer
256 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 $(\...
2
votes
0answers
89 views
Borel measurability
Suppose we have two locally compact Hausdorff spaces $X$ and $Y$. Let $i:X\to Y$ be a continuous injection. Under what condition the Borel $\sigma$-algebra of $X$ and $i(X)$ are isomorphic via the map ...
