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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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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 $$\text{supp}(\mu):=\{x\in\mathbb{R}^d:\text{every open neighbourhood of }x\text{ has positive measure}\}. $$...
A. Bond's user avatar
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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:...
f yz's user avatar
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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 ...
Shujun Tan's user avatar
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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 ...
Ogglie Ostrich's user avatar
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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 ...
Isochron's user avatar
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Image of measurable sets under one to one (a.e) functions

I want to know about this question that is image of a measurable set under a one to one (almost everywhere) function, measurable? Consider the following: Let $(X, \mathcal{B}_X)$ and $(Y, \mathcal{B}...
Reza Yaghmaeian's user avatar
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Let $\omega =\rho d\theta$ be a volue form of the circle $S^1$. Who are the diffeomorphisms of the circle that let $\omega$ invariant?

Consider the parametrization $\phi:]0,2\pi[\to S^1$ given by $\phi(\theta)=e^{i\theta}$. So the Lebesgue measure is given in local coordinates by the form $d\theta_z(\partial_z)=1$. I know that the ...
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Measure on the $d$-dimensional torus

I am looking for references, measure-integration theory where the $d$-dimensional torus $\mathbb{T}^d$ is treared rigorously: borel $\sigma$-algebra, measure functions, measures on $(\mathbb{T}^d,\...
mathex's user avatar
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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 ...
Sam's user avatar
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Prove that infinite union of $\mathcal{F_{i}}$ is not always $\sigma$-algebra

$\mathcal{F_{1}}$ is some sub-algebra and $\mathcal{F_{n+1}}$ is class of all sets that can be represented as a countable union or intersection of sets $\mathcal{F_{n}}$ Prove that $\bigcup_{n \in \...
Ryuk's user avatar
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Example of an infinite compact measurable space

Let $X$ be a nonempty set with a $\sigma$-algebra $\mathcal{A}$. The notion of $\sigma$-algebra strictly lies between Boolean algebras and complete Boolean algebras. Clearly, $\mathcal{A}$ is a ...
Dots_and_Arrows's user avatar
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20 views

Existence of weakly continuous regular conditional probability

Let $X$,$A$ be compact metric spaces. We consider Borel probability measures. Let $\mu$ be a probability measure on $X\times A$. Let $\hat{\mu}$ be the marginal of $\mu$ on $X$, i.e., for every Borel ...
Doug's user avatar
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Convergence on Borel sets implies weak convergence

Let $X$ be a locally compact space and let $M(X)$ denote all $\mathbb{C}$-valued regular Borel measures on $X$. Let $(\mu_n)_{n\in \mathbb{N}} \subset M(X)$ be a sequence. I want to show that $L(\mu_n)...
fish_monster's user avatar
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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. ...
reyna's user avatar
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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 ...
Thomas Winckelman's user avatar
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Definition of "interval of continuity" for function defined on sets

At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
Greg Martin's user avatar
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Is every Borel probability measure on Lindel$ӧ$f space $\tau$-additive?

It is well-known that every Borel measure on hereditarily Lindel$ӧ$f space X is $\tau$-additive, since we can extract countable subcover of every open cover of every subset of X and then apply $\sigma$...
Nina Badulina's user avatar
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Show that $\int_{\mathbb R^d}fd\mu=\sum_{x\in C}f(x)$ for countable $C\subset R^d$

Problem. Let $C$ be a countable subset of $\mathbb R^d$ and $\mu$ be the counting measure on $\mathbb R^d$ i.e., $$\mu(A)=\#(A\cap C),\qquad A\in \mathcal B(\mathbb R^d).$$ Show that for a measurable $...
reyna's user avatar
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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 ...
Beerus's user avatar
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Unique decomposition of complex Borel measure

I need to prove that the following decomposition of complex Borel measure is unique. $$\mu=\mu_d+\mu_{a c}+\mu_{sc}$$ Here $\mu_d$=discerete measure, $\mu_{ac}$=absolutely continuous measure, $\mu_{sc}...
Siddharth Prakash's user avatar
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Equivalence between Borel regular outer measure and regular measures

I am self learning some measure theory. Some sources like Evan's "Measure Theory and Fine Properties of Functions" define a Borel regular measure on a topological space $X$ as an outer ...
gordta_chichrron's user avatar
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1 answer
48 views

$\mathcal{F}_n$ measurable

Let $(X_n)$ be a sequence of random variables and $(\mathcal{F}_n)$ is associated naturel filtration. We define for each $k,n$ $S_{n,k}:= S_{n,k}(X_0,X_1,...,X_n)$ a function that depend on $X_0,X_1,.....
20Xblog8x12's user avatar
2 votes
2 answers
45 views

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 ...
Emerick's user avatar
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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 ...
Saúl RM's user avatar
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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 ...
Hoan Nguyễn's user avatar
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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 ...
Pallav Pant's user avatar
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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 ...
Steve's user avatar
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3 votes
1 answer
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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 ...
Saúl RM's user avatar
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Show that if $f$ is right continuous then $f$ is measurable

I'm trying to solve the following problem: Show that if the function $f: \mathbb{R} \to \mathbb{R}$ is right continuous then $f$ is measurable. (A function $f$ is measurable if $f^{-1}(E) \in \mathcal{...
MC2's user avatar
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An integral of a Borel measurable function w.r.t. Borel measure and that w.r.t. Lebesgue measure coincide with each other?

Here, I write $\mu$ as the Borel measure on $\mathbb{R^d}$ satisfying $\mu([a_1, b_1] \times \cdots \times [a_d, b_d]) = (b_1 - a_1) \cdots (b_d - a_d)$ ($a_i, b_i \in \mathbb{R}$) and $\lambda$ as ...
kuHamrry's user avatar
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1 answer
20 views

Apparent contradiction on Lebesgue-Stietjes measure?

I was reviewing Folland's Real Analysis. Given the Algebra of unions of disjoint half-open intervals (a,b]. And given an increasing, right continuous $F$. $\mu\left(\bigcup_1^n(a_j,b_j] \right) = \...
user258607's user avatar
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1 answer
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Folland chapter 3 exercise 22 Hardy-Littlewood maximal function and maximal theorem

Following is a question and it's solution from folland's real analysis chapter 3 exercise 22. Q22. If $f \in L^1\left(\mathbb{R}^n\right), f \neq 0$, there exist $C, R>0$ such that $H f(x) \geqq C||...
Siddharth Prakash's user avatar
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1 answer
129 views

Equal pushforward measures imply existence of random variable copy with matching joint?

Consider $\mathbb{R}^n$ along with its Borel $\sigma$-algebra, and let $\mathbb{P}$ be a non-atomic probability measure on $\mathbb{R}^n$. Let $f, g: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be two ...
Vokram8's user avatar
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1 answer
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Separability of codomains of Borel functions taking values in completely regular spaces

I am looking for a reference (or a counterexample) to the following statement. Let $X$ be a separable metric space. Suppose that $Y$ is a completely regular topological space and $f\colon X\to Y$ is a ...
Tomasz Kania's user avatar
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Explanation related outer measure and measurable cover

Theorem C is from the book Measure Theory by Halmos...from Chapter 3. My question: how Sigma finite of E is used in the proof? It will be very much helpful kindly give some hint or explanation.
Bikhu's user avatar
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1 answer
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On the right inverse of a Borel measurable map between two Souslin spaces being Souslin measurable, i.e. mapping open sets back to analytic sets

I'm currently reading some stuff on the existence of a measurable inverse, or measurable choice theorems from Bogachev's Measure Theory book, Volume II, where I'm trying to connect these two theorems ...
Learning Math's user avatar
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Lebesgue measurement exercise together with Borel algebra

I am having difficulty proving that the lebesgue measure $\lambda$ in the question below is the only one that satisfies the productory property. I imagine I need to consider an arbitrary measure $\mu$ ...
Andre Luiz's user avatar
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A bounded measurable function

Let $g:R→R$ be a non-negative integrable function. Let $f:R→R$ be a bounded measurable function satisfying $f(x)>1$ for every $x∈R$. Suppose that $∫_R f^n g≤M$ for every $n∈N$. Show that $g(x)=0$ ...
user1281744's user avatar
3 votes
2 answers
102 views

Example of Borel measure on R which is not Borel regular, but have finite value on all compact sets?

The answer below this question: Example of a Borel measure, which is not Borel-regular provides an example of Borel-irregular measure. Here, I am asking a harder question: Can we find a Borel measure $...
Ma Joad's user avatar
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2 votes
1 answer
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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 $\mu$ ...
Kiliroy's user avatar
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0 votes
1 answer
143 views

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 ...
Mr. Proof's user avatar
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1 vote
1 answer
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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 ...
Marta Sánchez Pavón's user avatar
1 vote
0 answers
55 views

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 (...
Frido's user avatar
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1 answer
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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 ...
S.H.W's user avatar
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1 vote
1 answer
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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^...
Anacardium's user avatar
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0 votes
1 answer
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Constructing a Radon measure from the integral of a function w.r.t another Radon measure

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 ...
Nazono Sumiko's user avatar

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