# Questions tagged [borel-measures]

Use this tag for questions related to Borel measures, which, on a topological space, are measures defined on all open sets.

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### Any set of $R^d$ is $G/delta$ or $F/sigma$ [closed]

is this true? We have to try it and we don't know where to start.
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If $X$ is a compact Hausdorff space, we know that finite Baire measures uniquely extend to a Radon measure. Furthermore every Radon measure also restricts to a Baire measure. We also know that every ...
1 vote
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### $\mu_0$ (semi-finite) is outer regular on all borel with finite measure

Let $\mu$ be a Radon Measure on $X$ and let $\mu_0$ be the semifinite part of $\mu$ ($\mu_0(K) = \sup \{\mu(F), F \subset K , \mu(F) < \infty \})$ a) Show that $\mu_0$ is inner regular in all ...
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### Subset $V$ Lebesgue measurable, but not Borel measurable

Define the function $g: [0, 1] \rightarrow [0, 1]$ with $$g(y) := \inf\{x \in [0, 1] : f(x) = y\},$$ where $f$ is the Cantor function. Now let $V \subset [0, 1]$ be a set that is not Lebesgue ...
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### Equivalence of Assertions for Functions of Bounded Variation

I am trying to prove that the following three assertions about a function $f$ on an interval $[a, b]$ are equivalent, but I am encountering some difficulties: $f$ is the difference of two increasing ...
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### Question on application of Borel-Cantelli lemma

I have an exercise in my measure theory class which goes as following: If $(X,A,\mu)$, where $A$ is a $\sigma$-algebra, is a measure space and $(B_n),n=1,2,\ldots$ a sequence of sets in $A$. Show that ...
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### Show that the counting measure $\zeta$ on $\mathbb{R}^{N}$ is Borel regular. [closed]

I want to show that the counting measure $\zeta$ on $\mathbb{R}^{N}$ is Borel regular. The counting measure is defined as follows: In addition, I know that the counting measure $\zeta$ is not $\sigma$...
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### Equality of Integrals implies equality of measures

I have some trouble solving the following problem: Let $\mu, \nu$ be two measures on the Borel-$\sigma$-algebra on $\mathbb{R}$. Assume that $\mu(\mathbb{R}) = 1$. Prove that, if for all continuous, ...
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### Why inner regularity of measure is defined as to be approximable from within by compact sets and not by closed sets?

This question has been asked here, but I don't believe that we have had a satisfactory answer, so I would like to reformulate that question. Please forgive me if this question turns out to be rather ...
1 vote
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### Definition of Borel regular outer measure.

I have ran across this in the hypothesis of one certain theorem. Let $X$ be a separable metric space with a Borel regular outer measure $\mu^{*}$ such that $\mu^{*}X = 1$. I don't understand the ...
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### Empirical distribution learns w.r.t total variation distance

I am trying to prove or disprove that the empirical distribution can learn any continuous distribution w.r.t the total variation distance. The context is the one of statistical learning. I am quite ...
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### Understanding proof of "if $f$ is non-negative, then it is the supremum of simple functions."

I am studying measure theory based on the book "Probability and Measure Theory" by Ash and have a question about understanding one of the theorems. Here it goes: (Page 43) Theorem 1.5.9 (d) ...
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### Correct terminology about "probability measures" and "probability distributions"

INTRODUCTION. Let's consider some previous answers (from previous questions) about "probability measure" and "probablity distributions": (Previous question 1) Distinguishing ...
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