Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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) $

3
votes
1answer
22 views

Let $m^\ast(E)$ < ∞.If for every interval (a, b) we have that $b-a$=$m^\ast((a,b)∩E) + m^\ast((a,b) ∩ E^c)$ then $E$ is lebesgue measurable

Since we are considering only few one type if sets (i.e open intervals) , i don't know how to prove $m^\ast(A)$=$m^\ast((A∩E) + m^\ast(A∩ E^c)$ using only information that $b-a$=$m^\ast((a,b)∩E) + m^\...
0
votes
1answer
13 views

outer measure on R some sort of continuity about Measure

$m^*(E)=q>0$,for any $c\in (0,q)$,there exist $E_0\subset E$,such that $m^*(E_0)=c$ $$m^*(E)=inf\{mG|E\subset G ,\text{G is open set}\}$$ I think if I can find a set A$\subset E$,and $m^*(E-A)=c$,...
0
votes
0answers
24 views

Proof of translation and linear transformation invariance of R_k

Let A be a Lebesgue measurable set. Suppose that $f$ and $g_r$ are respectively the translation and linear tansformation with ratio $r$ on $\mathbb{R}_k$. Prove that $f(A)$ and $g_r(A)$ are Lebesgue ...
1
vote
1answer
23 views

Example of countably additive function from a Boolean algebra to [0,infinity] that is not finitely additive

This example is implied to exist by Tao’s undergrad measure theory text, exercise 1.7.4, with the claim that further assuming that the empty set getting mapped to 0 prevents such a function from ...
1
vote
2answers
23 views

$ \inf_{ \text {open sets covering E} } ( \sum \text{length}(I_n) ) \leq \inf_{ \text {closed sets covering E} } ( \sum \text{length}(J_n) ) $

Let $E$ be a subset of $\mathbb R $, $J_n$ be a closed cover of $E$, where $\forall n, J_n = [a_n,b_n]$ and we build the open cover $I_n$ like this : $ \forall n, I_n = ]a_n - \frac {\epsilon }{2^{n+1}...
1
vote
0answers
24 views

Weaken the condition of measurability

Let $\mathcal{E}$ be a collection of subset of $\Omega$ and $\varnothing \in \mathcal{E},$ let $\mu$ be a countable subadditive set function defined on $\mathcal{E}$ for which $\mu(\varnothing) = 0.$ ...
0
votes
0answers
19 views

Why do I need this outer measure to be regular?

This is a problem in Halmos (it isn't my homework) that I have a proof for, but even though I can't identify the flaw in it, I know it must be wrong. Theorem: If $u$ is a regular outer measure on a ...
0
votes
2answers
72 views

Can a non-measurable set be measurable? (Seriously)

Perhaps I don’t speak English as well as I thought I did. In Folland, and other sources, I have encountered the following definitions: A “measurable space” consists of a set X together with a sigma-...
0
votes
0answers
37 views

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 $\varepsilon$ ...
0
votes
0answers
33 views

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)-...
0
votes
1answer
31 views

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 ...
0
votes
1answer
30 views

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{...
0
votes
0answers
68 views

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]) = ...
0
votes
0answers
8 views

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....
1
vote
1answer
62 views

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 ...
0
votes
0answers
31 views

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 ...
0
votes
0answers
14 views

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^*(\...
0
votes
1answer
33 views

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? ...
3
votes
1answer
45 views

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 ...
1
vote
0answers
40 views

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 ...
0
votes
1answer
46 views

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?
1
vote
1answer
36 views

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 \...
1
vote
1answer
36 views

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^*(...
0
votes
1answer
15 views

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^...
0
votes
1answer
20 views

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 $A$.
1
vote
1answer
20 views

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.
2
votes
1answer
45 views

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 $\mu^*$ ...
0
votes
1answer
62 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 $...
2
votes
1answer
34 views

$E \subset X$ 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^*(...
2
votes
1answer
74 views

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....
0
votes
2answers
45 views

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 ...
1
vote
1answer
26 views

$\{µ(E) : E ∈ S\} = [0, 1] ∪ [3, c]$. 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 ...
1
vote
0answers
29 views

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 ...
0
votes
0answers
17 views

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 ...
0
votes
0answers
71 views

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 ...
0
votes
1answer
36 views

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) = ...
5
votes
2answers
175 views

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 $m(A)=m(A\cap E) + m(A\cap E^c)$. ...
0
votes
1answer
598 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\{\...
1
vote
1answer
39 views

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, ...
1
vote
0answers
24 views

$|E|_e^{'} = |E|_e$ 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 ...
1
vote
0answers
87 views

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 $m_*(E)$ ...
2
votes
0answers
118 views

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 ...
4
votes
2answers
178 views

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\...
0
votes
1answer
32 views

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)&...
0
votes
0answers
29 views

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?
1
vote
0answers
24 views

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 ...
1
vote
1answer
116 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 ...
0
votes
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 ...
2
votes
1answer
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, ...
3
votes
0answers
34 views

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}\...