Questions tagged [measure-theory]
Questions about abstract measure and Lebesgue integral theory. Also concerns such properties as measurability of maps and sets.
793
questions with no upvoted or accepted answers
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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 $\mu$ ...
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213
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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 ...
2
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62
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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 $\...
2
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44
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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) ...
2
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63
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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,...
2
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159
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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 ...
2
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224
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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 ...
2
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55
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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 $X_1,\ldots,X_k\in(L^0)^d=\prod_{k=1}^dL^0$...
2
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283
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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 ...
2
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152
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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 ...
2
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81
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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 ...
2
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69
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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 $, $\alpha_{M}(x) = max_{|u| \leq M} |\phi(x,u)| \in L^{2}_{loc} (\Omega)$ .
...
2
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203
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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 ...
2
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217
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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}\...
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120
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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 $\sum X_i^2 = n$ ...
2
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58
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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 ...
2
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178
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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\...
2
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169
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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 $Y$ ...
2
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188
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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) \...
2
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160
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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+ \...
2
votes
0
answers
412
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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 ...
2
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answers
98
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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 ...
2
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0
answers
159
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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)$...
2
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59
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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 ...
2
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46
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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 ...
2
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136
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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|$,
$$
\...
2
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answers
244
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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 ...
2
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212
views
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 $$L(d,h)=\lim_{\...
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260
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Dual of $L^2(0,T,C)$ where $C'=BD(\Omega)$
Here is the hypothesis of my problem : $T>0$, $\Omega$ is a bounded open subset of $\mathbb{R}^n$ with a regular boundary. I'm actually looking for a proof that explains what is the dual of $L^2(0,...
2
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254
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questions about the Riemann-Lebesgue integral
(Assume Countable Choice.)
Generalizing arxiv.org/pdf/math/9903103 and www.um.es/beca/papers/BirkhoffBourgain.pdf,
the Riemann-Lebesgue integral can be defined as follows:
For $V$ a real $(\text{T}...
2
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0
answers
140
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Products for probability theory using zero sets instead of open sets
(For all of this post, at least Countable Choice is assumed to hold.)
For all Tychonoff spaces $\langle X,\mathcal{T}\hspace{.06 in}\rangle$ :
Define $\mathbf{Z}(\langle X,\mathcal{T}\hspace{.06 in}\...
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answers
125
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Problem of moments. Some generalisation.
Let $g(x) \colon \mathbb{R}^n_+ \to \mathbb{R}_+$ be homogeneous of order 1, concave and smooth function. Using entrywise product notation $x \circ p = (x_1 p_1, ..., x_n p_n)$ one can write
$$
\...
2
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0
answers
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 ...
2
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297
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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}^...
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 ...
2
votes
1
answer
610
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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
$$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &...
2
votes
2
answers
136
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Completion of the set of subsets with half volume.
Let $X$ be a measure space with finite $|X|=\int_X1$ and $f:X\rightarrow \mathbb{R}$ be a function. Under what condition on $X$ and $f$ does there exist a subset $Y \subset X$ satisfying the following?...
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30
views
Deterministic multifractal measure with quadratic singular spectrum?
For a non-negative locally finite measure $\mu$ on a bounded metric space $(\Omega,\mathcal{B})$, its local Holder exponent $f(x)$ is defined as $$f(x)=\lim_{r\downarrow 0}\frac{\mu(B(x,r))}{\log r}$$
...
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105
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Density of absolutely continuous measures on a Polish Space
Consider the set of all probability measures on a Polish space $X$ (equipped with the Borel $\sigma$-field $\mathcal{B}(X)$). I am wondering if there exist conditions under which a subset of measures ...
1
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0
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79
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Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
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 $\...
1
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0
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95
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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 $\...
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0
answers
85
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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 ...
1
vote
0
answers
75
views
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 ...
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 ...
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 ...
1
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0
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47
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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 ...
1
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0
answers
145
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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 ...
1
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0
answers
131
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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 $\...
1
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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}(...
1
vote
0
answers
62
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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 ...