# 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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### Modifiying a sequence of measures to assign a certain value when integrating a fixed function?

Let $f:\mathbb R ^d\to \mathbb R$ be some continuous function, $|f|\leq A(1+|x|)$, where $|\cdot|$ denotes the usual Euclidean norm. Fix a measure $\mu$ and constant $C$. Assume that $\mu_n$ is a ...
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### Sparse "bijection-proof" subsets of $[\mathbb{N}]^2$

We call a collection ${\cal S}\subseteq {\cal P}(\newcommand{\N}{\mathbb{N}}\N)$ bijection-proof if for any bijection $\varphi:\N\to\N$ there is $T\in{\cal S}$ with $\varphi(T) \in {\cal S}$. For any ...
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### Shrinking and expanding pairs in bijections $\varphi:\mathbb{N}\to\mathbb{N}$

Motivation. If we consider any bijection $\varphi:\newcommand{\N}{\mathbb{N}} \N \to \N$, we say integers $m\neq n$ are shrinking with respect to $\varphi$ if $|m-n|>|\varphi(m) - \varphi(n)|$, and ...
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### How to check that the surface measure is the weak limit of $\delta^{-1}\mathcal{L}^n|_{B(0,1+\delta)\setminus B(0,1)}$?

We denote the unit sphere $\{x\in\mathbb{R}^n:|x|=1\}$ by $S^{n-1}.$ If $x\in\mathbb{R}^n\setminus\{0\}$, the polar coordinates of $x$ are \begin{align*} r=|x|\in(0,\infty),\quad \gamma=\dfrac x{|...
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### Is there a name for finite unions of intervals?

Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
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### Sufficient conditions for the space of Radon measure to be a Banach space

Let $\mathcal{X}$ be a Hausdorff space and consider the space of Radon measures with bounded total variation $M(\mathcal{X})$ on $\mathcal{X}$. Usually, the additional assumptions on $\mathcal{X}$ are ...
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### How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?

The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
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### Topological measure theory on spaces that are not completely regular

In the usual discourse regarding approaches to measure theory, it is often pointed out that the restriction of topological measure theory to locally compact Hausdorff spaces is insufficient. However, ...
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### Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
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### When are increasing functions on posets (specifically, lattices) the CDF of a probability measure?

This is perhaps a basic question, but I couldn't find a reference. Let $P = (X,\leq)$ be a poset. Given a probability measure $\mu$ on $P$ (with respect to the Borel $\sigma$-algebra generated by sets ...
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### De la Vallée Poussin criterion on uniform integrability for infinite measures

The de la Vallée Poussin criterion (which is often used in combination with the Dunford-Pettis theorem) is usually formulated for probability measures/finite measures, for example in [Bogachev: ...
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Let $(\Omega,\mathcal{F},\mu)$ be a measure space (say complete and $\sigma$-finite, for simplicity). Furthermore, let $(X,\Vert\cdot\Vert_{X})$ be an arbitrary Banach space. I denote by $(L^{p}(\... • 1,289 2 votes 1 answer 121 views ### Domain of the infinitesimal generator of a composition$C_0$-semigroup In the paper [1] the following$C_0$-group is presented, $$T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E$$ where$E$is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ... 0 votes 0 answers 27 views ### Why$d\mu (q)\delta (k,q)$is$G$-invariant? Let$G$be a Lie group acting transitively on a smooth manifold$M$endowed with a quasi-invariant measure$\mu$(then there exists Radon-Nikodym derivative$\rho_f$for every$f\in G$). For$k\in M$,... • 287 0 votes 0 answers 66 views ### Measurable Extension Let$(\Omega, \mathcal{F})$be a measurable space and$X$some metric space (probably Polish) with the Borel$\sigma$-algebra and a function$f: \Omega \times X \to \mathbb{R}$. Usually, functions ... • 136 0 votes 1 answer 75 views ### Projection on a countable union of linear subspace For any natural number$n$,$V_n$denotes a closed linear subspace of a$L_2(m)$space, which is an Hilbert Space, where$m$denotes a finite measure. Moreover$(V_n)$is increasing, that is$V_n$is ... 0 votes 2 answers 137 views ### Is a signed measure$\mu$on$\mathbb{R}^d$characterized by the transform$\mathcal{L}_\mu (\lambda ):=\int e^{\langle \lambda,x\rangle }\mu (dx)$? In the book "Probability Theory" by Achim Klenke there's the following theorem: a finite measure$\mu$on$[0,\infty )$is characterized by its Laplace transform$\mathcal{L}_\mu(\lambda):=\...
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Let $E$ be a set, and let $S$ be any set of subsets of $E$, such that $S$ contains the empty set. Can you identify the smallest set of subsets $T$ of $E$ such that $T$ contains all elements of $S$, ...