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

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The counting measure on $\mathbb{R}^n$ is a map that takes a subset $A$ of $\mathbb{R}^n$ and returns its cardinality if it is finite or the symbol $\infty$ if it is infinite. So, if $A\subseteq\...
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The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference? Given a positive random ...
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I have a Markov kernel $(t,A) \mapsto \mathbb{Q}_t(A)$ from a standard Borel space $(T, \mathcal{T})$ into another standard Borel space $(\Omega, \mathcal{F})$. Also, for $t \neq s$, $\mathbb{Q}_t \...
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Let $(X,\mathcal X)$ and $(Y,\mathcal Y)$ be measurable spaces, $\pi \colon \mathcal Y\times X \to [0,1]$ a Markov kernel. We assume that it is measurably dominated, i.e. there is a $\sigma$-finite ...
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A metric space $(X,d)$ satisfies the Hoffmann-Jørgensen (HJ) property if for any two Borel measures $\mu_1,\mu_2$ we have that $\mu_1(B_r(x))=\mu_2(B_r(x))$ for all $r>0$ and $x\in X$ implies $\...
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On a measurable space $(E,\mathcal E)$, a stochastic kernel is a function $p\colon E\times \mathcal E\to [0,1]$ such that: for each $x\in E$, the function $A\mapsto p(x,A)$ is a probability measure; ...
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Let $\nu$ be a probability measure equivalent to $\mathbf{1}_{\mathbb{R}_+}(y) \, \lambda(dy)$. Let $\pi$ be a probability measure on $\mathbb{R}^2$ of second marginal $\nu$, such that $\nu(dy)$-a.e., ...
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Let $\mu$ be a finite measure on some measurable space $(X, \Sigma)$ and consider the topological vector space $L^0(\mu)$ of all real-valued measurable functions on $X$ with respect to convergence in ...
iolo's user avatar
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Consider a measure space $(S,\mu)$ and assume that $\mu(S)=1$. We consider the quantile function (or nonincreasing rearrangement) of a real valued function $f:S\to\mathbb{R}$ as the function \begin{...
Daan's user avatar
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Is there a formula $\phi$ in the language of set theory such that $$ \text{ZFC proves } \exists x \in \mathbb{R}:\text{ the set }A_x​:=\{y\in\mathbb{R}:\phi(x,y)\} \text{ is not Lebesgue measurable?} $...
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In teaching multivariable Riemann integration, I was trying to develop the theory of successive Riemann integrals (so all start with the one-dimensional case familiar to the students) as far as ...
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I am really wondering how to prove this lemma from the book 'Counting processes and survival analysis'. No need for the first and second point, just the third point, why does the maximum random ...
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I'm interested in Freiling's axiom of symmetry and I specifically wonder if it may be proven from more basic axioms about measures on $\mathbb R^n$, in the sense that there is a sequence of measures $\...
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Theorem 15.1 in Classical Descriptive Set Theory by Kechris states: (i) Let $X, Y$ be Polish spaces and $f:X\rightarrow Y$ be continuous. If $A\subseteq X$ is Borel and $f|_A$ is injective, then $f(A)$...
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Suppose we have two measure spaces, $(\Omega, \mathscr{F}, \mu)$ and $(\Psi, \mathscr{G}, \nu)$, with $\mu(\Omega) = \nu(\Psi) < \infty$, and we consider the set of measure isomorphisms mod 0 ...
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Analytic sets are projections of Borel sets, and are known to be Lebesgue measurable (in fact universally measurable). The question of whether measurability of analytic sets can be shown in some ...
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Let $\mathfrak{L}$ denote the $\sigma$-algebra of Lebesgue-measurable subsets of $\mathbb{R}$. Consider a null subset $\mathcal{N} \in \mathfrak{L}$. Does the subset of $\mathbb{R}^{2}$ $$ \bigcup_{t \...
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I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
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Let $\mathcal{E}$ be the class of Lebesgue-measurable subsets of $\mathbb{R}$. The notion of $(\mathcal{E},\mathcal{E})$-measurable function is a bit pathological since lots of continuous functions ...
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Let $(X, \mathcal{B})$ be a compact metric space, and let $(\mu_n)_{n\ge 1}$ be a sequence of probability measures on $X$ such that $\mu_n$ converge weakly towards $\mu$. Let $f : X \to \mathbb{R}$ be ...
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Let $X$ be a locally convex space, let $(p_t)_{t\in T}$ be a net of Borel probability measures on $(X,\sigma(X,X^*))$, and let $p$ be a $\tau$-additive (in particular, Radon) with respect to the weak ...
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Suppose $G_1,G_2$ are compact $T_2$ groups (if it helps, we can assume that $G_1=G_2^n$ for some $n\in\mathbf N$) and suppose $B\subseteq G_1\times G_2$ is Borel (if it helps, we can assume that $B$ ...
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Let $X$ be a topological space (or metric space if needed) and let $(p_t)_{t\in T}$ be a net of Borel probability measures on $X$ which converges weakly to a Borel probability measure $p$, that is, ...
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Let $U \subseteq \mathbb R^n$ be an open set. The Banach space $ba(U)$ is the space of bounded finitely additive signed measures on the Borel $\sigma$-algebra of $U$. Its norm is the variation. $ba(U)$...
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I have a question about Bochner spaces, especially valued in Lebesgue $L^p$ spaces. I have been reading Analysis in Banach Spaces Volume I by Tuomas Hytönen, Jan van Neerven, Mark Veraar and Lutz ...
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Noam Elkies maintains a page of mathematical miscellany on his website. The last entry on this page is a problem he proposed to the American Mathematical Monthly, rejected on the advice of both ...
Will Brian's user avatar
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Setting Let $f\ge 0$ be a measurable function on $[0,\infty)$ and define $g(s,t)=ste^{-s-st}$. For $\lambda>0$ set $$ I(f,\lambda)=\int_0^\infty g(s,f(\lambda s))ds. $$ Let $\mu_T$ be the ...
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Let $(X, \mathcal{B}, \mu)$ be an arbitrary measure space and $T: X \to X$ a measure-preserving transformation. Fix a measurable set $A \in \mathcal{B}$ with $0 < \mu(A) < \infty$, and fix $t \...
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Let $(X,d,\mu)$ be a metric measure space. Does there exist $f\in L^1(X,\mathbb R)$ so that $$\mu\left(\left\{x\in X:\lim_{r\to 0^+}\frac{\int_{B_r(x)}f(y)d\mu(y)}{\int_{B_r(x)}d\mu(y)}= f(x)\right\}\...
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For a $\sigma$-ideal $\mathcal{J}$ on $\mathbb{R}$, consider the following statement. There is a partition $\bigsqcup_{i < \kappa} A_i = \mathbb{R}$ such that $\mathcal{I} = \{X \subseteq \kappa: \...
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There is an absolutely continuous version of the measure theory statement of Leibniz's rule (see https://math.stackexchange.com/questions/1683350/differentiability-under-the-integral-sign-of-...
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Motivation: As a consequence of this question, every function $u$ in $W^{1,2}(\mathbb{R}^n,\mathbb{R})$ has a representative $u^*$ such that there is a null set $E\subset \mathbb{R}^n$ with the ...
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I am looking for good references (survey, monograph, or paper with a solid background section) on the Banach space / functional analytic structure of spaces of finite signed measures $\mathcal{M}(X)$, ...
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Let $(\Omega,\mathcal{F},\mu)$ be a measure space and consider a function $f \colon \mathbb{R} \times \Omega \to \mathbb{R}.$ For the problem I work on, a seemingly good hypothesis to place on $f$ is ...
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I am studying the problem of representing positive linear functionals on the space of bounded norm-continuous functions on a separable infinite-dimensional Hilbert space $H$. A known challenge is that ...
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Suppose $X$, $Y$ are probability spaces and $K$ is a Markov kernel from $X$ to $Y$. Is it the case that $K$ induces a positive unital normal map $T_K : L^\infty(Y) \to L^\infty(X)$ given by $T_K(f)(x) ...
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Let $G$ be an $n$-dimensional Lie group. Then there is a Haar measure $\mu$ and an invariant metric $d$ on $G$. If $G=\mathbb R^n$ and $d$ is the Euclidean distance, then we have that $$\lim_{r\to 0^+}...
user479223's user avatar
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Let $H$ be a separable infinite-dimensional Hilbert space over $\mathbb{R}$ with norm $\|\cdot\|$. Let $\{\mu_\alpha\}$ be a net of Borel probability measures on $H$, and let $\mu$ be a Borel ...
Zlyp's user avatar
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Suppose we have two Markov kernels $(A, t) \mapsto K(X \in A \mid T=t)$ and $(A, t) \mapsto H(X \in A \mid T = t)$ that represent conditional distributions of $X$ given $T=t$. We obtain the marginal ...
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$\DeclareMathOperator\supp{supp}\newcommand\teq{\underset t=}\newcommand\tlt{\underset t<}$Let $(X, \mathcal{B}(X),\mu)$ be a probability space, where $\mathcal{B}(X)$ is the Borel $\sigma$-algebra ...
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Jerison and Kenig give the following two conditions (alongside an exterior corkscrew condition, which is not relevant to this discussion) in the definition of an $(M,r_0)$ nontangentially accessible ...
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Suppose $P$ is a finitely additive probability measure on a space $\Omega$, and $X_1,X_2,X_3,\dots$ are i.i.d. random variables with mean 0 and variance 1. Is it true that for all $r \in \mathbb{R}$, ...
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Suppose $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space. Let $f:\mathbb{R}^n \to \mathbb{R}$ be a real-valued function, $\varphi:\mathbb{R}^n \to \mathsf{P}(\mathbb{R}^n)$ be a set-valued ...
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Let $\mathbb{N}$ be the set of positive integers. For $n\in\mathbb{N}$, let $\sigma(n) = \sum\{k\in\mathbb{N}: k<n \land k|n\}$. Let $C$ be the closure of the set $\{\sigma(n)/n:n\in\mathbb{N}\}$ ...
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It is a result of Levin and (independently?) Kautz that if $X \in 2^\omega$ is 1-ML random relative to a computable measure $\mu$ the either $X$ is computable (it is an atom of $\mu$) or $X$ is Turing ...
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In the construction of the Haar measure on a locally compact Hausdorff group $G$ that is standard in the literature, one usually makes a combinatorial definition first: Given a compact set $A$ and an ...
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Note: We denote by $\mu$ be the usual Lebesgue measure on $\mathbb R$. For $E$ a Lebesgue measurable subset of $\mathbb R$, we define its lower asymptotic density at $x \in \mathbb R$ by $$\liminf_{r \...
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In the usual framework for conditional expectation we consider a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and a numerical random variable $X$ on $\Omega$, i.e., a random variable that ...
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Let $(\Omega, \mathcal{A})$ and $(\mathcal{X}, \mathcal{F})$ be measurable spaces. And let $K:\Omega\times\mathcal{F}\rightarrow [0,1]$ be a markov kernel. I wanted to know if it is possible to ...
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Let $(X,Y)$ be a pair of random variables with joint distribution $\rho$ and marginals $\alpha$ and $\beta$. We define the operator the conditional expectation $$ S: L^1(\beta) \to L^1(\alpha) $$ by $$...
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