# 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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### Convergence of total variation given uniform convergence of variation in open sets

Let $\Theta$ be a metric space, and let $\{\mu_\theta\}_{\theta \in \Theta}$ be a collection of probability measures over the Borel $\sigma$-algebra $\mathcal B(X)$ of some metric space $(X, \rho)$. ...
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### Is a Radon measure always positive on non-empty open sets?

A measure is locally finite if it is finite on all compact sets from the underlying $\sigma$-algebra. A measure is regular if every measurable set $A\in\Sigma$ can be approximated from above by open ...
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### Prove continuity of a function constructed from a Borel measure

If $\mu$ is a finite Borel measure on $\mathbb{R}$, then the function $F^{\mu}: \mathbb{R} \rightarrow \mathbb{R}$ , $F^{\mu}(\lambda) = \mu((-\infty, \lambda])$ is right-continuous. Attempt at ...
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### Borel measure that is locally finite but not borel regular

I was wondering if there is an obvious example of a borel measure $\mu$ on $\mathbb{R}^{n}$ that is locally finite but not borel regular, i.e. if there exists $E\subset\mathbb{R}^{n}$, such that for ...
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### How to prove that the dual of $L^\infty$ is the set of bounded finitely additive measures?

How to prove that the dual of $L^\infty(X,\mu)$ is the space of finitely additive, signed measures on $X$ that are absolutely continuous with respect to $\mu$? I only find that it's a "well known ...
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### Does the set of open sets in $\mathbb{R}$ is countable?

I want to prove that the cardinality of Borel $\sigma$ algebra is $\aleph$, using the next proposition: If $E \subset P(X)$ is infinite, and the cardinality of E is $\aleph$ , the $\sigma$ algebra ...
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### If $f(x)$ is a Borel function, is $f(x+a)$ also a Borel function?

I've been recently covering Borel functions and wondering are they preserved by the following property. If $f(x)$ is a Borel function then $f(x+a)$ is also a Borel function where $a$ is a real number. ...
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### Show that monotone function f: I->R is measuable on an interval

Show that a monotone function f: I -> R is borel-measureable. I know, that this question has been asked and answered multiple times here, but only for f: R->R und not for f: I->R. The first ...
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### Show that two measures are the same

I have that $u$ and $v$ are two measures on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$ and $u=((-\infty,a]) = v((-\infty,a]) < \infty$ for all $a\in \mathbb{R}$ and I want to show that $u=v$. I know ...
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### Understanding the definition of convolution in Bauer's book

Am trying to understand the the definition of convolution in Bauer's measure theory book. He writes: Let $\mathcal{M}^d$ denote the set of all finite Borel measures on $\mathcal{B}^d$. Initially we ...
Given a complete separable metric space $(X,d)$. My question is if it is true that for any finite positive Borel measure $\mu\in\mathcal{M}^+_b(X)$ there exists a sequence of finite atomic measures $... 1answer 112 views ### Show that$\nu(E) = \int_E \phi \,d \mu$is inner and outer regular. Let$\phi \geq 0$be a function in$L^1(\mu)$where$\mu$is a Radon measure (= a Borel measure on$X$that is finite on compact sets, inner regular on open sets and outer regular on all compact sets) ... 1answer 32 views ### Folland exercise: measure on the countable ordinals I managed to prove$(a),(b),(c),(d),(e)$from the following exercise in Folland's book: I'm currently attempting$(f)$. In particular, I am stuck at showing that$\mu$is countably additive. So let$(...
Give an example of a Borel measurable function f from R to R such that there does not exist a set B ⊂ R such that $| R \setminus B |$ = 0 and $f|_B$ is a continuous function on B. By Lusin's theorem ...