# 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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### Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?

The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete? Indexing any countable set, ...
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### Equivalence between notions of dynamical coupling as defined by Villani in his book Optimal Transportation: Old and New

$\DeclareMathOperator\law{law}$In Villani's book he presents the following notions of dynamical couplings: Let $(X,d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure ...
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### Set where the speed of convergence is uniform in Lebesgue's density theorem

Let $B \subset \mathbb R^n$ be the unit ball. Consider a Borel measurable set $E \subset B$ with positive Lebesgue measure $|E|>0$ (say $|E| = |B|/2$). Then, Lebesgue's density theorem, says that ...
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### Well-definedness of maximum likelihood estimation

Consider a family $\{\mu_\theta:\theta\in\Theta\}$ of probability measures on a measurable space $X$. Given $x\in X$, the maximum likelihood estimate is the value of $\theta$ which maximizes the ...
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### Confusion around uniform integrability and Vitali convergence theorem

Motivation The notion of uniform integrability is important for formulating the Vitali convergence theorem. Unfortunately, different authors define uniform integrability differently, which causes ...
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### A MNC with maximum property but not singular

Let $E$ be a Banach space, $\mathfrak{M}_E$ indicate the family of all nonempty bounded subset of $E$, $\mathfrak{N}_E$ the family of all relatively compact sets, and $Ker \mu=\{X\in \mathfrak{M}_E$ ...
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### Supremum with respect to the order of measures on $(X,A)$

Suppose that $(X,\leq )$ is an ordered set, we can define the maximum and the infimum of this set,now let $(X,A)$ be a measurable space and let $M(X,A)$ be the set of all measures on $(X,A)$, we now ...
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### Does $\mathbb R^n$ equipped with a sum of Dirac delta measures admit nowhere locally constant continuous integrable functions?

For any point $x \in \mathbb R^n$, denote by $\delta_x$ the Dirac Delta measure centered at $x$. Let $a_n$ be an sequence of positive numbers with $\lim_{n \to \infty} a_n = 0$, and let $d_i$ be a ...
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### Space of functions and the Coordinate process

I have the following question: Let a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a stochastic process $X$ be given and define $(\mathcal{F}_t) = \sigma(X_t;t \geq 0).$ Further, we have ...
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### Is the singular value decomposition a measurable function?

$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators $$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$ where $\mathbb U_n$ is the ...
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### When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist?

When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist ? Here $\mu$ can be any measure (Lebesgue, Borel, Haar etc), $f$ is a measurable function, $S$ is any measurable set with ...
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### What is the 'right' definition of zero measure subsets of Banach spaces?

Question. There are several ways of defining a notion of a 'zero measure' subset of a Banach space $X$. Which one is the 'right' or failing that, the preferred notion? [See below for a more precise ...
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### A strong Borel selection theorem for equivalence relations

In Kechris' book "Classical Descriptive Set Theory" there is the following theorem (12.16): Let $X$ be a Polish space and $E$ an equivalence relation such that every equivalence class is ...
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### Averaging and fractional Laplacian

Let $u,\phi:\mathbb R \to \mathbb R$ be smooth functions and $\Omega_\epsilon$ be a bounded domain in $\mathbb R$ with diameter $\epsilon>0$ (consider for exaple the ball $B_{\epsilon/2}(0)$). Is ...
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### Reference request: large generalized probability measures

I'm interested in references relevant to the following: what is the right generalization, if there is one, of a probability measure that takes on values in an structure of more than continuum size? I'...
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### (Generalized) Uncentered Maximal Function $\tilde Mf$ in Stein's Harmonic Analysis

It is well known that on $\Bbb R^n$, equipped with the usual Lebesgue measure, the standard Hardy-Littlewood maximal function $Mf(x)$ (with respect to averaging on cubes or balls centered at $x$) is ...
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### Subspace of RKHS generated by kernel mean embeddings

Suppose $\mathcal{H_k}$ is a reproducing kernel Hilbert space (RKHS) with reproducing kernel $k: \mathcal{X} \times \mathcal{X} \rightarrow \mathbb{R}$. I am looking for results characterising the ...
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### "Orthogonalization" of bilinear operator

I would like to ask if the following fact correct/known: Let $D$ is dyadic grid, $ֿ\mu$ and $\nu$ are positive measures, $\{\alpha_I : I \in D\}$ is set of positive numbers indexed by cubes in the ...