# Tagged Questions

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

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### Is every regular Borel outer measure topologically additive?

If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive? If so what is a proof or a counterexample? Definitions: Topologically Additive: $X$ is a topological space, $m$ ...
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### Random walk on d-dimensional torus

I am reading the following paper: http://arxiv.org/pdf/1602.03849v2.pdf I will explain the general setup below. Let $x\in X=\mathbb{T}^d$, where $\mathbb{T}^d$ is the d dimensional torus. Let $\rho$ ...
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### Random Variables with Simply Connected Support [closed]

Let $X$ be a random variable with simply connected support on real line. Define $Z_n=\sum_{i=1}^n X_i$ and $X_i\sim X$ for all $i$. Does $Z_n$ has simply connected support, for all $n\geq 1$?
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Consider the following situation: Let $\Omega =l^{\infty}(\mathbb{R})$ be the space of all bounded sequences of real numbers. We will consider in $\Omega$ the metric: $d(x,y)=\sum_{i\geq 1}\frac{|... 1answer 96 views ### Some integrals with respect to a Gaussian measure on a Hilbert space Assume that$(H,\langle\cdot,\cdot\rangle)$is a real separable Hilbert space equipped with a Gaussian measure$\mu$with a mean$m$and a covariance operator$C$. Let$x\in H$be a fixed vector. What ... 0answers 99 views ### Are the theorems of Ergodic Theory valid for non-probability spaces? The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ... 4answers 504 views ### Speed of convergence in Lebesgue's density theorem Let$\lambda=\text{unif}([0,1])$be uniform distribution on$[0,1]$and$B$be any Borel set. Lebesgue's density theorem states that for$\lambda$-almost all$x\in[0,1]$the limit $$\lim_{\epsilon\... 1answer 334 views ### Does there exist a continuous measure'' on a metric space? Let X be a separable complete metrizable space. Does there exist a complete metric d and a Borel measure \mu such that (a) \mu(B_r(x))<\infty for every open ball B_r(x) of radius r>... 0answers 77 views ### Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set? I also put this question on MSE here Let \Gamma\subset \Omega\subset \mathbb R^N be such that \mathcal H^{N-1}(\Gamma)<+\infty (this also implise that \Gamma is Hausdorff measurable). Let \... 0answers 67 views ### Characterizing the optimimum over the space of probability measures Consider the following optimization problem: $$\max_{\mu \in \mathcal{M}} \int \log\left( \int e^{\alpha U(x,y)} d\mu(y) \right) d\nu(x)$$ where \mathcal{M} is the space ... 1answer 98 views ### Results for Hausdorff Measure after Linear Transformation For the Sierpinski Triangle, S, the d dimensional Hausdorff measure is given by, H^{d}(S). If a linear transformation, W is applied to S, with$$W(x,y)=\begin{bmatrix} 1/2 & 0 \\ 0 &... 1answer 66 views ### Usable Change-of-Variables Formula for Hausdorff Measure Let$H^{s}$be the$s$-dimensional Hausdorff measure, let$D$be a nonsingular matrix. Consider the change of measure formula: $$\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)... 0answers 44 views ### 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... 1answer 76 views ### sequence of graphs converge in the sense of varifold to multiplicity 2 plane Say in R^3, is there a sequence of smooth graphs f_n over some plane P, such that the graphs as submanifolds in R^3 converge in the sense of varifold (as Radon measures on R^3 \times Gr(2,3) ) ... 2answers 146 views ### How many cuts are required for a weighted-proportional cake-cutting? In proportional cake-cutting, there are n agents with equal entitlements to a "cake" (an interval). Each agent i has a nonatomic value measure V_i over the cake, and it is required to create a ... 0answers 70 views ### Tensor product of sigma-algebra? Does there exist an explaination of the product of 2 sigma-algebra in terms of of tensor product ? Which could explain the tensor product sign for the product of 2 sigma-algebra. In particuliar, does ... 0answers 179 views ### Topology on the space of Borel measures Let B be the set of all measures \phi of \mathbf{R}^{n} such that every open set is \phi -measurable (sometimes these measures are called Borel measures). Note the measures in B are ... 1answer 121 views ### \nu is a Dirac delta Let X be an locally compact Hausdorff space and m a positive regular Borel probability measure where m(Y) is 0 or 1 for any Borel set of X. Does it necessarily follow that m is a Dirac delta?... 0answers 66 views ### Reference for a simple fact in measure theory (semi-algebras) What is the textbook where the following simple fact can be found: A measure defined on a semi-algebra S can be extended to a sigma-algebra generated by S. In the texbooks that I have looked into ... 1answer 76 views ### Metrizability of the space of probability measures endowed with the topology of setwise convergence Let X be a separable completely metrizable space, let \mathscr{B}(X) denote the Borel \sigma-algebra on X, and let \mathscr{P}(X) denote the space of all probability measures on (X, \... 0answers 47 views ### The density one properties of \mathcal H^{N-1}-rectifiable set Let S\subset \mathbb R^N be a \mathcal H^{N-1}-rectifiable set. Then we know that there exist countably many Lipschitz N-1-graphs \Gamma_i\subset \mathbb R^N such that$$ \mathcal H^{N-1}\... 2answers 234 views ### Can a nowhere differentiable function preserve measurability? I want to know whether a continuous nowhere differentiable function$f: \mathbf{R} \to \mathbf{R}$can map Lebesgue measurable sets to Lebesgue measurable sets. More generally I'm interested to know ... 0answers 58 views ### Converge of measures [closed] Good night, we have the following: Let$(X,d)$is a general metric space,$\mathcal{M}(Y)$is the set of finite Borel measures on$Y$and$C_B(Y)$denotes the Banach space of bounded continuous ... 1answer 187 views ### Product of limit$\sigma$-algebras Let$X$and$Y$be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space$Z$, let$\mathscr{S}(Z)$denote the limit$\sigma$-algebra on$Z$, i.e. the smallest$\...
Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...