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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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Subset of Lebesgue measurable subset of Vitali set is NOT Lebesgue measurable

$B_x = \{y \in [0,1]: x-y \in \Bbb{Q}\}$, $ \varepsilon=\{C \subset [0,1]: \exists x \quad C=B_x\} $. By the axiom of choice we can choose exactly one element of each equivalence class $\varepsilon$ ...
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Prove $A\subseteq B \subseteq \mathbb{R}\implies m^*(B)-m^*(A)\leq m^*(B \setminus A)$

Prove $$A\subseteq B \subseteq \mathbb{R}\implies m^*(B)-m^*(A)\leq m^*(B \setminus A)$$ where $m^*(B)$ is finite Let $s_i\subseteq t_i$ Be the covers of $A\subseteq B\ $ respectively $1$.$m^*(B)-...
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Lebesgue outer measure of image of set is less than or equal to the Lebesgue outer measure of set

For a differentiable function $f : \mathbb{R} \to \mathbb{R}$ with $|f'(x)|\leq 1$ and any set $E \subset \mathbb{R}$ $$m^*(f(E)) \leq m^*(E)$$ where $m^*$ is the outer Lebesgue measure First, the ...
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Show example $E \subset \mathbb{R}^n$ where exterior measure does not equal that of the smallest open ball cover.

Let $m^*$ be Lebesgue outer measure (also called exterior measure) on $\mathbb{R}^n$. Suppose $E$ is a subset of $\mathbb{R}^n$ with $m^*(E) < \infty$. Let $\mathcal{O}_m$ be the open set: \begin{...
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Lebesgue Outer Measure Limit

Let $\mu^{*}$ be the Lebesgue outer measure on $\mathbb{R}$. I found the following exercise in a textbook. Exercise. For every $A \subseteq \mathbb{R}$, $\lim_{k \to \infty} \mu^{*}(A \cap [-k,k]) = ...
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Is the Caratheodory $\sigma$-algebra generated by a semi-ring $R$ only dependent on $\sigma(R)$?

Let $\mu$ be a measure defined on a certain $\sigma$-algebra $\Sigma$, and let $R,Q\subseteq \Sigma$ be two different semi-rings of sets within it, such that their generated $\sigma$-algebras coincide....
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What is the most elegant known proof of $m^*(A) \leq \sum_{n} m^*(A_n)$ when $A = \bigcup_n A_n$?

Let $A = \bigcup_{n \in I} A_n \subset \Bbb{R}^k$ where $I$ is an arbitrary index set. Define the Lebesgue outer measure by $m^*(A) := \inf \ \{ \sum_{n} \text{vol}(I_n) : I_n, n \geq 1$ are each ...
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Complete Measure spaces

Let $\mu^*$ be an outer measure and $\mu$ be a measure on $(\Omega,\mathcal F)$. $\mu^*$ can measure any subset of $\Omega$ while $\mu$ can only measure the sets in $\mathcal F$. Let's take some set ...
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Proving outer measure is additive over disjoint open cells (without Caratheodory)

Is there a way to prove that outer measure is additive over disjoint open cells without Caratheodory's condition/theorem? Ie, if the $E_k$ are mutually disjoint open cells, I want to prove that $m^*(\...
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Are Cat. II sets measurable (for a given outer measure)

Given a complete metric space $X$. Let $\mu^*(E)=0$ if $E$ is of Cat. I, $\mu^*(E)=1$ if $E$ is of Cat. II. I have proved that $\mu^*$ is an outer measure. What are the $\mu^*$-measurable sets? ...
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Is there an outer measure on $\mathbb R$ whose only measurable sets are $\mathbb R, \emptyset$?

I.e. an outer measure $\mu^*$ on the reals such that if for some $A \subset \mathbb R$, we have that for all $B \subset R$ $\mu^*(B) = \mu^*(B \cap A) + \mu^*(B \setminus A)$, then $A=\mathbb R$ or $A ...
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Example of measure for some algebra over $\mathbb N$

$\mathcal F$ is a set of events ($\sigma$-algebra). Can you give an example for some algebra $\mathcal A$ over $\mathbb N$ a non-zero finite additive measure $\mu $ on this algebra, which has a ...
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If $A, B \in [m,n] \subseteq \mathbb{R}$ are sets of measure zero, then is $AB = \{ab | a \in A, b \in B\}$ measure zero? [closed]

If $A, B \in [m,n]$, where $[m,n] \subseteq \mathbb{R}$ are sets of measure zero, then is $AB = \{ab \; | \; a \in A, b \in B\}$ measure zero?
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Is my Proof Correct: $(X, \mathcal{A}, \mu)$ $\sigma-$finite measure space then $\mu^{*}(B) < \infty$

My proof is different to my professor's and I wanted to ensure that mine is still correct and I am not overlooking something. So: Let $\mu$ be a $\sigma-$finite measure on $X$. Show that for $B \in \...
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Continuity from below of outer measure extending an algebrea

Consider the following problem: Let $\mathcal{A}$ be an Algebra with an additive function $\mu$ and let $\mu^*$ be the outer measure corresponding to $\mu$. Show that if $A_n\uparrow A$ then $\mu^*(...
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Finite outer measure $\mu$ on a particular set implies the existence of another set such that outer measure and measure are equal on respective sets

Let $(X, \mathcal{A}, \mu)$ be a measurable space with a $\sigma-$finite measure $\mu$, while $(X, \mathcal{A^{*}}, \mu^{*})$ is the completion of the underlying space. Show: For $B \in \mathcal{A^...
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What is the relationship between $m^*(A-B)$ and $m^*(A)-m^*(A\cap B)$

What is the relationship between $m^*(A-B)$ and $m^*(A)-m^*(A\cap B)$. Here $m^*(A)$ means the exterior measaure of $A$.
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Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable?

Does there exists a set which has the exterior measure,but doesn't Lebesgue measurable? Can one give a example of this ? I do really puzzled.
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Premeasures on Algebras

I have a question that I can't seem to work out. Specifically, I don't entirely understand what is being asked. The questions is: Let $\mu_0$ be a premeasure on an algebra $\mathcal{A}$. Let $\mu^*$ ...
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56 views

Prove: $m^*(aE+b)=|a|m^*(E)$

For all $E\subseteq \mathbb{R} $ and $a,b\in \mathbb{R}$ we define $aE+B:=\{ax+b:x\in E\}$ Prove: $$m^*(aE+b)=|a|m^*(E)$$ I know the following properties: $m^*(\emptyset)=0$ and so does for $...
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$E \subset X$ and $\mu ^* (E)=0$, then $E$ is $\mu^*$measurable

Let $\mu^*: P(X) \rightarrow [0, +\infty]$ be an outer measure and $E \subseteq X$such that $\mu^*(E)=0$. I want to show that $E$ is $\mu^*$measurable. For this we have to show that: $$\mu^*(A)=\mu^*(...
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If a sequence of measurable functions $f_{n}$ converge to $f$ almost everywhere then $f$ is measurable

$\textbf{Theorem}$ if a sequence of measurable functions $f_{n}$ converge to $f$ almost everywhere then $f$ is measurable $\textbf{proof}$ Let $A=\{ x\in X : \lim f_n(x)=f(x)\}$. Since $f_n \to f$ a....
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Lebesgue outer measure equivalence to another set function on compact sets

Denote the Lebesgue outer measure $m^*(A) = inf \{\sum_{n=1}^{\infty} \ell (I_n): A \subseteq \cup_{n=1}^{\infty} I_n$ where $I_n$’s open intervals$\}$. Consider the “measure” (which is, as it turns ...
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$\{µ(E) : E ∈ S\} = [0, 1] ∪ [3, c]$. Prove that $c ≥ 4$. Can you give an example of $(X, S, µ)$ if $c = 4$?

Suppose that there exists a measure space $(X, S, µ)$ with $\{µ(E) : E ∈ S\} = [0, 1] ∪ [3, c]$. Prove that $c ≥ 4$. Can you give an example of $(X, S, µ)$ if $c = 4$? My Try: Clearly $\mu(X)=c$ Now ...
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Does any set of representative rationalis have known inner and outer measures?

For any $x,y\in I=[0,1]$, define $x\sim y$ iff $x-y\in\mathbb Q$. Then $I$ is a disjoint union of equivalence classes of $\sim$. By the axiom of choice, one can form a set $V$ by choosing one ...
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Inner and outer measures on sets of type $\cal{G}_\delta$ and $\cal{F}_\sigma$

I would like to prove a certain inequality involving the inner and outer Lebesgue measure and the existence of sets $K \in \cal{F}_\sigma$, $H \in \cal{G}_\delta$ with $K \subset E \subset H$, for an ...
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Outer and inner measures

I want to prove that $m_{∗}(A \cap (a, b)) = (b − a) − m^* (A^c \cap (a, b))$. Since $(a,b)$ is open, it is measurable. Then $(b-a)=m^*((a,b))=m_*((a,b)$. Also by the definition of measurable ...
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Showing that the set of $\mu^*$-measurable sets in a $\sigma$-finite measure space is of a certain form.

Suppose we have a $\sigma$-finite measure (i.e. $E$ is a countable union of finite measure sets) space $(E,\mathcal{E},\mu)$. Say a set is $N$ is null if $N \subseteq B \in \mathcal{E}$ with $\mu(B) = ...
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Is the Carathéodory measurability criterion optimal in some sense?

If $m$ is an outer measure defined on a set $X$, we say that a subset $E$ of $X$ is Carathéodory-measurable with respect to $m$ if for all subsets $A$ of $X$, we have $m(A)=m(A\cap E) + m(A\cap E^c)$. ...
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360 views

Prove that every open set is lebesgue-measurable

Let $\mathbb{R}^n\supset{I}=(a,b)=(a_1,b_1)\times...\times(a_n,b_n)$ with $a,b\in{\mathbb{R}^n}$ such that $a_i\lt b_i $$\forall i$. So that the outer measure is defined as: $$ \mu^*(A)=inf\{\...
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Given $E$ in $\mathbb R^n$, there exists an open $G$ containing $E$ such that $m^*(G)\le m^*(E) + \epsilon$. But $m^*(G-E) < \epsilon$ may not hold?

We let $m^*(E)$ denote the outer measure of a set $E$ in $\mathbb R^n$. Given $E$ in $\mathbb R^n$, there exists an open $G$ containing $E$ such that $m^*(G)\le m^*(E) + \epsilon$. In general, ...
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$|E|_e^{'} = |E|_e$ for every $E \subset \mathbb R^n$, where $|E|_e$ is the exterior measure of the set $E$

Consider a fixed rotation of the usual coordinate system in $\mathbb R^n$. Notions pertaining to the rotated system will be denoted by primes. Thus, $I'$ denotes an interval with edges parallel to the ...
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Prove that a subset of $\mathbb{R}$ with positive outer measure contains almost a whole interval

This is exercise number 28 from chapter 1 of Stein & Shakarchi's Princeton Lectures in Analysis III: Real Analysis. Let $E$ be a subset of $\mathbb{R}$ with $m_*(E) > 0$, where $m_*(E)$ ...
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What is the inutition behind the Caratheodory criterion for measurable sets? [duplicate]

Before the criterion of Caratheodory a set $A$ was defined to be measurable iff the inner and outer measure of it coincide $\lambda_*(A)=\lambda^*(A)$. Caratheodory found a definition for measurable ...
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Why is the inner measure problematic?

Currently the Lebesgue measure is defined by the outer measure $\lambda^*(A)$ by the criterion of Carathéodory: A set $A$ is Lebesgue measurable iff for every set $B$ we have $\lambda^*(B)=\lambda^*(B\...
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Outer measure inequality

I came upon this statement in a proof I read today and I can't figure out, why the following inequality (at $(*)$) holds: Let $\nu$ be outer measure on a set $X$, let $\mathcal M:=\{A\subset X|\nu(A)&...
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Premeasure and induced outer measure exercise.

I'm solving the following exercise from Folland's Real Analysis book: I have done everything except item (c), I can't find a way to prove it, any hints?
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About alternative definition of $s$-dimensional Hausdorff outer measure in $\Bbb R^n$

I have this definition for the $s$-dimensional outer Hausdorff measure for some subset $A$ of a metric space $(X,d)$: $$\mathcal H_*^s(A):=\lim_{\epsilon\to 0^+}\mathcal H_\epsilon^s(A)\tag1$$ where ...
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1answer
83 views

Lebesgue outer measure of the union is strictly less than the sum of the measures

Show that it exists $A , B \subset \mathbb{R}^n$ such that $A \cap B = \emptyset $ and \begin{align*} \lambda^{*} (A \cup B) < \lambda^{*} (A) + \lambda^{*} (B) \end{align*} I tried to use ...
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1answer
25 views

Outer measure of the set $A= \lbrace x \in \mathbb{R}^n :\exists r>0 \; with \; \mu(B_r(x))=0 \rbrace$

id like to show if i have a outer measure $\mu$ and the set $A= \lbrace x \in \mathbb{R}^n :\exists r>0 \; with \; \mu(B_r(x))=0 \rbrace$ than $\mu(A) = 0$. I don't know how to start. I need ...
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41 views

Showing $m^*(A-E)=m^*(A)$ when $m^*(E)=0$

I'm trying to show that $m^*(A-E)=m^*(A)$ when $m^*(E)=0$ where $m^*$ is the outer measure. So my plan is to show that $m^*(A-E) \geq m^*(A)$ and $m^*(A-E) \leq m^*(A)$ (and this one is easy, ...
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16 views

Sum of two additive set functions and Caratheodory measurability

Let $\frak{R}$ be a ring of subsets of $X$, and $\mu,\nu$ be two finitely additive $[0,\infty]$-valued maps on $\frak{R}$. As usual, $\mu$ and $\nu$ give rises to outer measures $\mu^*$ and $\nu^*$, ...
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Outer measure induced by a Jump function

This is from exercise 4.4 of Elstrodt's measure theory textbook. By a jump function $F:\mathbb{R}\rightarrow\mathbb{R}$ we mean a function which can be written in the form $$F(x)= \begin{cases}\...
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1answer
18 views

For every r>0, there is a vitali set having outer measure less than r

For every $r>0$, there is a vitali set having outer measure less than $r$. Here is my approach: Take arbitrary positive real number r. The interval $(0,r)$ has positive outer measure and hence ...
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31 views

Condition for algebra being sigma algebra

Suppose that we defined $\mathcal{M}_{\mu^{*}}=\left\{B:B \ \ \mu^{*}-\text{measurable}\right\}$ as the set of all $\mu^{*}$ measurable sets.A set is measurable with respect to outter measure $\mu^{...
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1answer
57 views

Show that a certain closed interval does not contain an open interval $I \neq \emptyset$.

The question: Let us enumerate the set of rationals $\mathbb Q = \{r_n\}_{n=1}^{\infty}$ and define \begin{equation} B = \bigcup_{n=1}^{\infty} \left(r_n - \frac 1 {2^n}, r_n + \frac 1 {2^n}\right)\,....
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28 views

prove of two sets being equal

I am trying to show two sets are equal. Basically I have a set that is outer measurable, i.e. it satisfies the following: $m^{*}(A) \geq m^{*}(A \cap E) + m^{*}(A \cap E^c)$ where $E$ is a ...
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1answer
47 views

If $A \subseteq \mathbb{R}$ satisfies $m^\ast(A) = 0$, then there exist $B, C ∈ \mathcal{B}(\mathbb{R})$ such that $A = B \setminus C$?

Is this a true claim or false? Recall that $m$ is Lebesgue measure and $m^\ast$ is Lebesgue outer measure. Also, $\mathcal{B}(\mathbb{R})$ in this case represents the Borel $\sigma$-algebra on $\...
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24 views

Prove that for any $k \in \Bbb{R}$ we have$ \quad \mu^{*}(kA) = |k|\mu^{*}A, $

$A \in \Bbb{R}$. Prove that for any $k \in \Bbb{R}$ we have$ \quad \mu^{*}(kA) = |k|\mu^{*}A, $ where $kA:=\{ka: a \in A \}$ $\mu^{*}(kA)=inf |I_n|_{kA \subset \cup I_n}$ To bo honest I am not sure ...
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44 views

Finite sub-additivity of outer measure; is my proof correct?

I tried to come up with a proof on my own and I would appreciate it if someone could please verify my proof, give me any suggestions for it, and help me strenghten one step of it: Goal: $\text{ ...