# 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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### 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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### 9K Bartle $A\subset B$ with $l^*(B\setminus A)=0$

Let $A$ be a borel measurable set. I need to prove the existence of $B$ in the Borelians such that $A\subset B$ and $l^*(B\setminus A)=0$. Because of the way the outer measure $l^*(A)$ is defined (as ...
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### Example of a set mapping that is not an outer measure by violating either monotonicity or subadditivity

Let $X$ be a non-empty set and $\mu:2^X\to[0,\infty]$ be a map. Do you happen to know good examples of such $\mu$s that do not satisfy either the monotonicity or countable subadditivity of a proper ...
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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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### Does Carathéodory's extension of measure space preserve inclusion?

A measure space $(X,\mathcal{S},\mu)$ can be extended to a measure space $(X,\hat{\mathcal{S}},\hat{\mu})$ as follows: $\mu$ is extended to an outer measure $\mu^\ast$ (defined as the infimum of the ...
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### Measure spaces that stabilize with respect to the Carathéodory extension

When learning measure theory, for sure one encounters the Carathéodory extension theorem with extends a premeasure defined over a semiring to a measure defined over a $\sigma-$algebra. Let $X$ be a ...
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### Is there a nonmeasurable subset of $\mathbb{R}^2$ that is $1-$dimensional Hausdorff measurable?

For $n\in\mathbb{N}^*$ and $s\in\mathbb{R}_{\ge 0}$, the $s-$dimensional Hausdorff measure $H^s$ is an outer measure over $\mathbb{R}^n$, and the $\sigma-$algebra of $s-$dimensional measurable subsets ...
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### Can the measure restricted from an outer measure extend to a different outer measure?

Let $X$ be a set, and $\rho$ an outer measure on $X$. Let $\mathcal{M}(\rho)$ be the set of $\rho$-measurable sets. Then the restriction $\rho|_{\mathcal{M}(\rho)}$ is a measure on $\mathcal{M}(\rho)$....
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### Extending a finite measure from a semi-ring $\mathcal{E}$ to $\sigma(\mathcal{E})$

Let $\mathcal{E}$ be a semi-ring of subsets of $\Omega$ and let $\mu:\mathcal{E}\rightarrow [0,\infty]$ be a measure on $\mathcal{E}.$ Define the outer measure $\mu^{*}$ generated by the measure $\mu$ ...
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### Measure not equal to set function that generates outer-measure Counterexample

Let $M\subseteq 2^X$ be a collection of subsets of $X$ and $\rho:M\to [0,\infty]$ be a set function such that $\emptyset,X\in M$ and $\rho(\emptyset)=0$. The set function $\rho$ induces an outer ...
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### A lemma used to prove Besicovitch Theorem

Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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### A question about Theorem 2.8.2 of "Geometric Measure Theory" of Federer.

Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...