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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Why does $C=\{x\in\mathbb{R}:\nu(\{x\})\neq0\}$ belong to $\mathscr{B}(\mathbb{R})$?

Let $\nu$ be a finite measure on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$. Let $C=\{x\in\mathbb{R}:\nu(\{x\})\neq0\}$. I want to show that $\nu$ can be decomposed into the sum of a discrete meausre, a ...
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1 vote
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Compactly supported Borel measure $\mu$ on $\mathbb{R}^d$ [closed]

Definition : If $\mu$ is any Borel measure on $\mathbb{R}^d$ then its support is defined as $$\text{supp}(\mu):=\{x\in\mathbb{R}^d:\text{every open neighbourhood of }x\text{ has positive measure}\}.$$...
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Is the Evaluation Map on Bounded Borel Measurable Functions Borel Measurable?

I am working with a set $I$, defined as the closed interval $[0,1]$, and a set $X$, which consists of all bounded Borel measurable functions defined on $[0,1]$. The uniform metric on $X$ is defined by:...
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Why does an invariant measure define a Schwartzman cycle?

If $M$ is a manifold with a flow $\phi_X:\mathbb{R}\times M\rightarrow M$ induced by a vector field $X\in \Gamma TM$. Any Borel measure $\mu$ defines a $1$-current $c_\mu$, i.e. an element in the dual ...
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Borel $\sigma$-field and the product $\sigma$-field

I have a question about the proof of lemma 1.5 of the textbook Measure Theory, Probability, and Stochastic Processes by Le Gall. Lemma 1.5: Suppose that $E$ and $F$ are separable metric spaces, and ...
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Standard measure and Standard topology on the set $[0,\infty]$

I'm currently trying to follow along with some lecture notes on probability and measure theory, but I need some help understanding some definitions Let $f: E \rightarrow G$ be a measurable function ...
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Finding measurable subsets of any given value?

I'm not sure if the following is true, but I would hope it is with the regularity properties of $\mu$. Let $X$ be a locally compact Hausdorff space with $\mu$ a nonzero Radon measure on $X$. Then ...
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Existence of probability measures $P$ such that for any $x\in \Omega$ and closed set $x\not\in A$, the inequality $0<P(A^c)$ holds.

Suppose $(\Omega, \Sigma)$ is a measurable space such that $\Omega$ is a topological space and $\Sigma$ is a Borel $\sigma$-algebra on $\Omega$. For any such measurable spaces $(\Omega, \Sigma)$, does ...
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Does there always exist a probability measure $P$ such that for any $x,y\in X$ there exist neighborhoods $x\in U,y\in V$ for which $0<P(U),P(V)$?

Consider a measurable space $(X, \Sigma)$ such that $X$ is a Hausdorff space and $\Sigma$ is a Borel $\sigma$-algebra on $X$. For any such measurable space $(X, \Sigma)$, does there exist a ...
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When is measurability of a function $f$ equivalent to $f$ being almost everywhere the limit of measurable functions?

I have a suspicion that the following is true: $\def\AAA {\mathcal{A}} \def\BBB {\mathcal{B}} \def\NN {\mathbb{N}}$ Theorem: let $(X,\AAA,\mu)$ be a complete measure space, and $(Y,\BBB)$ a measurable ...
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Show that $\int_a^bF(x)dx+\int_{F(a)}^{F(b)}F^{-1}dy=bF(b)-aF(a)$ [duplicate]

Problem. Suppose $-\infty<a<b<\infty$ and $F:[a,b]\rightarrow R$ is strictly increasing and continuous. Prove that $$\int_a^bF(x)dx+\int_{F(a)}^{F(b)}F^{-1}(y)dy=bF(b)-aF(a).$$ Attempt. ...
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Which Probability Distributions Dominate the Lebesgue Measure?

Recall In probability theory, the distribution $\mu_X$ of a random variable $X$ (on some unspoken probability space) refers to the measure $\mu_X(A) := \mathbb{P}(X \in A)$ that is defined on the ...
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Borel Functions That Are Continuous

I am self-studying measure theory. I know that not all Borel functions are continuous. But I would like to explore certain conditions that make a Borel function continuous. In this post, we saw that ...
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An example of an $L_p$ space with Borel measure and dimension larger than $\mathfrak c$

Given a normed space $X$, we can define the Borel $\sigma$-algebra in $X$ as the smaller $\sigma$-algebra containing all the open (or closed) sets of $X$. Suppose a measure $\mu$ is chosen in the ...
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Can I suppose a measure preserving action is by homeos?

Let $G$ be a countable group with a measure preserving action on a probability measure space $(X,\mathcal{B},\mu)$. I am looking for sufficient conditions (as general as possible) so that there is a ...
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1 vote
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Can the support of a Borel probability measure has measure between 0 and 1?

On Wikipedia, the support of a Borel measure $\mu$ on a topological space $X$ is determined by: $$\operatorname{supp}\mu=\lbrace x\in X:\mu(U)>0,\forall U \text{ open, containing } x\rbrace.$$ The ...
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Limit of the measure of the delta neighborhood of Borel Sets.

Statement 1.7 from Fractal Geometry by Falconer I am reading through Fractal Geometry Mathematical Foundations and Analysis by Falconer and I am not very familiar with measure theory. While reading ...
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For a Borel subset $B$ of a complete, seperable metric space $S$ and $\epsilon > 0$, there exists compact $C \subset S$ with $P(B) < P(C) + \epsilon$.

For my bachelor thesis, I've been studying iterated random functions and a very limited amount of measure theory to understand it rigorously. One thing I could not understand is the following: Suppose ...
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Sufficient conditions for finitely supported measures being dense

Let $(X,\mathcal{B})$ be a Hausdorff topological space with its Borel $\sigma$-algebra. What are some general conditions we could impose on $X$ so that finitely supported measures (i.e. finite affine ...
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Is a measure, whose distributional derivative is a measure, absolutely continuous wrt Lebesgue?

Suppose $\mu$ is a finite Borel measure on $\mathbb{R}^n$ with the property that it's distributional gradient $\nabla\mu$ is a vector-valued finite Borel measure. Does it follows then that $\mu$ ...
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Prove $\nu$ is absolutely continuous to Lebesgue measure if and only if $f$ is absolutely continuous.

Let $\nu$ be a finite Borel measure on $[0,1]$. Define $f : [0,1] \to \mathbb R$ by $f(x) = \nu ([0,x))$. Prove $\nu$ is absolutely continuous to Lebesgue measure (\mu) if and only if $f$ is ...
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Example of a continuous measure which is mutually singular wrt Lebesgue measure

We are are asked to find an example of a measure $\lambda$ such that $$f(x)=\int_0^x d\lambda >0$$ for all $x>0$, $f$ is continuous in $[0,1]$, and $\lambda$ is mutually singular wrt the ...
1 vote
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Proof that continuous functions are Borel measurable from Axler, "Measure, Integration and Real Analysis"

I am going through Axler's "Measure, Integration and Real Analysis" and have a question about his proof that continuous functions are Borel measurable. There has been a question about this (...
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Constructing measure using a non-monotonic function

It's well-known that we can construct a measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$ using a Stieltjes measure function $F:\mathbb{R} \to \mathbb{R}$ which is a non-decreasing right continuous ...
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Difficulty in showing that $u$ is harmonic.
Let $\mu$ be a complex regular Borel measure on $\mathbb T.$ Define a function $u : \mathbb D \longrightarrow \mathbb C$ by u \left (re^{i \theta} \right ) = \int_{\mathbb T} \frac {1 - r^2} {1 + r^...
The problem is to show Suppose that $\mu$ is a Radon measure on $X$, If $\phi \in L^1(\mu)$ and $\phi \geq 0$,then prove that $\nu(E)=\int_E \phi d\mu$ is a Radon measure. My attempt: We first show ...