Questions tagged [measure-theory]

Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.

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Isomorphism of probability spaces

Consider a surjective map $f:(X, \sigma(f)) \to (Y, \mathcal{Y})$, if a measure $\nu$ is given in $(Y, \mathcal{Y})$ the pullback $\nu(f(\cdot))$ is a measure on $(X, \sigma(f))$, similarly if $\mu$ ...
Jorge E. Cardona's user avatar
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Question on application of Tonelli's theorem in convolution (topological groups)

Let $G$ be a locally compact group which is a countable union of compact subsets and $\lambda$ a left Haar measure on $G$ (i.e. $(G,\lambda)$ is $\sigma$-finite). Let $f,g \in L^1(G)$. Then I have a ...
TheGeekGreek's user avatar
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Extending a Baire-to-Baire-Kernel to a Borel-to-Borel Kernel

It is well known that a finite measure on the Baire $\sigma$-algebra of a, say, compact Hausdorff space can be extended to a unique regular measure defined on the Borel $\sigma$-algebra. The Baire $\...
Michael Greinecker's user avatar
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Maximizing the sum of a decreasing function over a separated set

Fix $d>0$. Let $f:[0,\infty)\to(0,\infty)$ be a decreasing function of $x$ for $x\geq d$. Let $S_d\subset\mathbb{R}^n$ represent a set of points containing the origin such that the (Euclidean) ...
brett1479's user avatar
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Sensitivity of a function against its random arguments

Let $g:R^{n+m} \to R$ be a deterministic function of some independent random variables $x_1,\ldots,x_n$ with distributions $f_{x_1}(x),\ldots,f_{x_n}(x)$ and some deterministic variables $z_1,\ldots,...
Jeff's user avatar
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A construction of abstract Wiener spaces using Prokhorov's theorem

I am struggling with Leonard Gross's (original) construction of abstract Wiener spaces (AWS). His proof is somewhat convoluted, but from what I have been able to understand he constructs a certain ...
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Characterizing the optimimum over the space of probability measures

Consider the following optimization problem: \begin{equation} \max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x) \end{equation} where $\mathcal{M}$ is the space ...
Adrien's user avatar
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Does there exist $\lambda_{\sigma(1)}$ such that $\mu(A\cap\{\lambda_{\sigma(1)}\neq0\})>0$?

Let $(\mathcal F,\Omega,\mu)$ be a measure space and $A\subseteq\Omega$ such that $\mu(A)>0$. Let $L^0$ be the space of all measurable functions. We say $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
user74301's user avatar
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Measure of the Attractor of Critical Points of a Manifold

Let $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a smooth function and consider the $d$-manifold $M = \{(x, f(x)): x \in \mathbb{R}^d\} \subset \mathbb{R}^{d+1}$. Let $P$ be a property of some subset ...
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Interchanging integrals and continuous linear forms in RKHS

I am reading Reproducing kernel Hilbert spaces in probability and statistics by A Berlinet, C Thomas-Agnan. In Chapter 5 INTEGRATION OF $\mathcal{H}$-VALUED RANDOM VARIABLES they write One of the ...
Manuel Schmidt's user avatar
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Uniform convergence of long geodesic to Liouville measure

Here is the set up : let $(S,g)$ an hyperbolic surface and $L_g$ the associated volume measure. By the shadowing lemma there exist sequences of long closed geodesics, $\gamma_n$ which approximate the ...
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Doubts regarding pre-compactness of bounded sequence of measure valued functions

Given: $\phi(x,\lambda) : \Omega$X$\mathbb R \to \mathbb R^{n}$ be a Caratheodory vector such that for each $M \gt 0 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ . ...
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On the use of the term "field of sets" in Maharam's papers

I am reading some papers by D. Maharam, and feel a little bit confused about her use of the term "field of sets". Nowadays, I think the term is standardly used to mean a pair $(X, \mathscr{F})$ for ...
Salvo Tringali's user avatar
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Stopping time sigma-fields

Let $(F_n)$ be a discrete Filtration and $S_n,S$ (not necessarily finite) stopping times with $S_n\uparrow S$ (increasing convergence). Is it true that the associated sigma-fields satisfy $F_{S_n}\...
nullptr's user avatar
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Coupling Marginals of Distributions on the Sphere

Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ ...
AustinC's user avatar
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A canonical example of the non-existence of predictive probability distribution

Section 3 of Fortini et al. (2000) states that Given $(X^\infty, \mathcal X^\infty,P)$, a predictive probability distribution of $x_n$ given $(x_1, \dots, x_{n-1})$ with respect to $P$ need not ...
Mohamad's user avatar
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Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?

Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable). Also, let $f:D_1\cup D_2=D\...
Elliot's user avatar
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Sequences of transition probability measures

Suppose that $X$ and $Y$ are compact metric spaces. A Borel probability measure $\mu$ on $X\times Y$ satisfies $$ \mu(A\times B)=\int_A\mu(B|x)\mu_X(dx), $$ for $A$ and $B$ Borel sets in $X$ and $Y$ ...
mo15's user avatar
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Progressive measurability and functional composition

Suppose we have a progressively measurable process $X$ taking values in $\mathbb{R}^d$. What are sufficient conditions on a function $$f( x, t, \omega ) \colon \mathbb{R}^d \times [0,\infty) \...
Tobi's user avatar
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Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?

Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed) taking values in $[-1,1]$ that have the following property: 1) The average $A_n := \frac{(X_1+ \...
Ritwik's user avatar
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Lebesgue point and regularity of functions

A known theorem says that for $f \in L_{loc}^1(\mathbb{R}^d)$, almost every point is a Lebesgue point. I know too a theorem saying that for $f \in W_{loc}^{1,p}(\mathbb{R}^d)$ , every point is a ...
user62319's user avatar
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Finding a general form of the density function when we have a four dimensional random variable

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
Sedi's user avatar
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Borel class of a set of measures

Let $K$ be a compact Hausdorff space and consider $X=Ball(M(K))$, the unit ball of the space of regular Borel measures on $K$. Endow $X$ with the weak-$*$ topology $\sigma(M(K),C(K))$, regarding $M(K)$...
Fred Dashiell's user avatar
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59 views

Stronger version of linearity for functions of measures

Let $X$ be a standard Borel space, and $P(X)$ be space of Borel probability measures on $X$. It is also a standard Borel space if endowed with the topology of weak convergence, so we can integrate ...
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Where to read about this kind of "measure of irredundancy" of a set from a family of sets?

Studying a very practical problem from psychometrics, I encountered the following construction. Let $(X,\mu)$ be a measure space; if preferred, you can presume $\mu$ is a probability measure. In any ...
მამუკა ჯიბლაძე's user avatar
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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 $\phi(x):=1+|x|$, $$ \...
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Packing symmetric matrices in spectral norm, and defining measures on symmetric matrices

I'm trying to upper bound the $\epsilon$-packing number of $\Theta=\{A\in\mathbb{S}^{d}:\; a\preceq A \preceq b\}$ (where $\mathbb{S}$ are symmetric $d\times d$ matrices) for some $a\leq b$ with ...
martin's user avatar
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Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define $$L(d,h)=\lim_{\...
Joseph Van Name's user avatar
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260 views

Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$

Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of $L^2(0,...
user37238's user avatar
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questions about the Riemann-Lebesgue integral

(Assume Countable Choice.) Generalizing arxiv.org/pdf/math/9903103 and www.um.es/beca/papers/BirkhoffBourgain.pdf, the Riemann-Lebesgue integral can be defined as follows: For $V$ a real $(\text{T}...
user avatar
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
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Problem of moments. Some generalisation.

Let $g(x) \colon \mathbb{R}^n_+ \to \mathbb{R}_+$ be homogeneous of order 1, concave and smooth function. Using entrywise product notation $x \circ p = (x_1 p_1, ..., x_n p_n)$ one can write $$ \...
Appliqué's user avatar
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668 views

Standard Borel Space

Let $(X,Z,\mu)$ be a standard borel measure, I will like to know if there is an isomorphism mod zero preserving the measures form $([0,1],B,\lambda)$ where $\lambda$ is absolutely continuous with ...
Carlos De la Mora's user avatar
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297 views

volume under induced metric of preimage set of a regular value under a polynomial map

I am interested in the following kind of polynomial map: $$ f: \mathbb{T}^k \to \mathbb{R}^n$$ where $f$ is a polynomial map of a certain maximum degree $d$, in the sense that if we imbed $\mathbb{T}^...
John Jiang's user avatar
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2 votes
1 answer
110 views

Closedness of the set of probability measures anihilating a measurable function

Let $X$ be a compact metrizable topological space and let $f$ be a bounded, real valued, Borel function on $X$. Denoting by $P(X)$ the collection of all probability measures on $X$, consider the ...
Ruy's user avatar
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1 answer
610 views

Results for Hausdorff Measure after Linear Transformation

For the Sierpinski Triangle, $S$, the $d$ dimensional Hausdorff measure is given by, $H^{d}(S)$. If a linear transformation, $W$ is applied to $S$, with $$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
Zachary W. Robertson's user avatar
2 votes
2 answers
136 views

Completion of the set of subsets with half volume.

Let $X$ be a measure space with finite $|X|=\int_X1$ and $f:X\rightarrow \mathbb{R}$ be a function. Under what condition on $X$ and $f$ does there exist a subset $Y \subset X$ satisfying the following?...
asmc's user avatar
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Deterministic multifractal measure with quadratic singular spectrum?

For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$ ...
MikeG's user avatar
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1 vote
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105 views

Density of absolutely continuous measures on a Polish Space

Consider the set of all probability measures on a Polish space $X$ (equipped with the Borel $\sigma$-field $\mathcal{B}(X)$). I am wondering if there exist conditions under which a subset of measures ...
d.k.o.'s user avatar
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Measurability of the union of cut loci along a curve

Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define $$ U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s)) $$ as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
Hengchao Chen's user avatar
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95 views

If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions Representation Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$. We call $\...
S-F's user avatar
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$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
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Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$

This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
David Gao's user avatar
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1 vote
1 answer
110 views

In the limit, do the intersection points of a string figure define a probability measure on the unit disk?

Let D = {z ∈ ℂ | |z| ≤ 1} denote the closed unit disk in the complex plane. For any integer n ≥ 1 define the nth string figure S(n) ⊂ D as the union of all n(n+1)/2 line segments that connect two ...
Daniel Asimov's user avatar
1 vote
0 answers
78 views

Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$

Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...
Daniel Goc's user avatar
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0 answers
47 views

Understanding simple point processes (part 2)

This is a follow up of this previous question. I'm trying to understand the following proposition from An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods by Daley ...
matteogost's user avatar
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145 views

Generalization of Borel functional calculus

[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus] Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
oggius's user avatar
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Optimal transport-like problem where the objective depends on conditional probability distribution

$\DeclareMathOperator\marg{marg}$I would like to know if the following problem can be studied as an optimal transport problem, possibly imposing additional assumptions on the data. Consider two sets $\...
Francesco Bilotta's user avatar
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0 answers
63 views

Physical measure of a dynamical system in terms of its density

Let $f$ be a $\mathcal{C}^1$ vector field on a compact subset $M \subset \mathbb{R}^n$. We define a dynamical system by $$\dot{x}(t)=f(x(t))$$ In ergodic theory, the occupation measure is $$\mu_{x, T}(...
NicAG's user avatar
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Benjamini-Schramm convergence: convergence on metric balls implies weak convergence?

In the famous paper by Benjamini-Schramm 2001, they consider the space of rooted graphs with uniformly bounded degrees, this space modulo rooted graph isomorphism is denoted by $\mathcal X$. This ...
Fredy's user avatar
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