Questions tagged [outer-measure]
Outer measure on $X$ is a function $ \phi : 2^X \rightarrow [0,\infty]$ defined on all subsets on $X$ that satisfies the following : (1) $\phi ( \emptyset )=0$ (2) Monotonicity : $A\subset B$ implies $\phi (A)\leq\phi (B)$ and (3) Countable subadditivity : $\phi (\bigcup_{i=1}^\infty A_j) \leq \sum_{j=1}^\infty \phi (A_j) $
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Extending a local property of a measure to a global property of the measure.
Let $\mu_1$ be an outer measure, $\mu_2$ be a measure, and $E$ be a bounded subset of $\mathbb{R}^d$. Suppose that for every $x\in \mathbb{R}^d$, there exists a neighborhood of $V_x$ of $x$ such that $...
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Lebesgue measure generating set
In my cource of measure theory, Lebesgue measure were built starting with measure on semiring $S\subset 2^X$, after we defined outer measure for each $A\in2^X$ by $\inf\{ \sum{m[B] : A\subset \cup B , ...
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Outer Measure in terms of disjoint elements
I have a question about outer measures. My book defines an outer measure by
$\mu(E)=\inf\{\sum m(A_n) : A_n\in\mathcal{R}, E⊂\bigcup A_n\}$
where $\mathcal{R}$ is a ring and $m$ a measure on it. Now ...
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Understanding infimum and jordan measure cover by open and closed set
I'm self-studying the Jordan measure and its definitions. I have a question regarding how the outer measure $ m_o(E) $ behaves when $ E $ is a closed interval and we restrict the elementary set to be ...
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Measure theory, prove the countable additivity of measure
I am reading Cohn's Measure Theory, here is an exercise from Chapter 1, Section 2.
Let ($X,\mathcal A,\mu$) be a measure space, and define $\mu^{\star}:\mathcal A\rightarrow[0,\infty)$ by
\begin{...
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Prove that there exists a bounded set $A\subset R$ such that $|F|\leq |A| − 1$for every closed set $F\subset A$. [duplicate]
Prove that there exists a bounded set $A\subset R$ such that $|F|\leq |A| − 1$ for every closed set $F\subset A$.
$|·|$ is outer measure. Below is my idea.
Let $A$ be $[0,1]-\mathbb{Q}$, so $|A|=1$. ...
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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 ...
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Prove that $\mu_*\leq\mu^*$ where the definition of an inner and outer measure is induced by a measure
I know that similar question has been asked and answered here, here, and here. But I am looking for a different proof based on different definitions.
We have the following definition of an outer and ...
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Composition of Lebesgue measurable function and Invertible linear transformation is Lebesgue measurable
Let $f:\mathbb{R}^n\to \mathbb{R} $ is a Lebesgue measurable function and $T\in Gl(n,\mathbb{R})$ then show that $f\circ T$ is also Lebesgue measurable .
For Borel measurable functions it's easy to ...
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Question About The Remark after Proposition 1.4.11 from Measure Theory by Donold Cohn
My Question
Define subsets $G$, $G_0$, and $G_1$ of $\mathbb{R}$ by
\begin{align*}
G &= \{x:x=r+n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\\
G_0 &= \{x:...
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Property of Outer Measure on $\mathbb{R}$
Question From - Axler Measure Theory - Problem 3 - Section 2A
Throughout: For $A \subset \mathbb{R},$ $|A|$ denotes the outer measure of $A$ and is defined
$|A|=inf\\{\sum_{k=1}^{\infty}\ell(I_k): I_1,...
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In the definition of outer measure, can we replace "open intervals" by "disjoint open intervals"
The definition of the outer measure of a set $A\subseteq\mathbb{R}$ is as follows:
$$
|A| = \inf \left\{ \Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k): I_1, I_2,\dots\text{ are open intervals such that }A\...
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Invariance in outer measures
Recently I was trying to solve this problem.
Suppose that $X$ is a set and $\mu^{\star}$ is an outer measure on $2^X$. Let $A \subseteq X$ be a set such that $\mu^{\star}(A)<+\infty$ and suppose ...
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Measurability of the set of elements who belong to a infinite amount of subsets in a sequence [closed]
I've been struggling to prove the following statement:
Let (X,$\mathcal{M}$,$\mu$) a finite measure space and let $(A_n)_{n\in\mathbb{N}}$ a sequence of measurable sets in X. Now consider $M$ the set ...
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Is there not translation invariant "measure"
I was reading Sheldon Axler's Real Analysis book and he mentions the following (as does the course I am taking) :
There does not exists a function $\mu : \mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$...
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