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) $

Filter by
Sorted by
Tagged with
0
votes
0answers
15 views

Expression of Lebesgue inner measure.

For bounded set $E \subset \mathbb{R^n},$ difine the Lebesgue inner measure $m_*$ as below. $m_*(E):=m^*(Q)-m^*(Q\cap E^c) \big(=v(Q)-m^*(Q\cap E^c)\big)$ where $Q$ is a closed hypercube in $\mathbb{R^...
0
votes
1answer
24 views

Proof from the definition that the outer measure of any open set is the sum over its interval lengths.

Let $|A|$ denote the outer-measure of a set of real numbers $A\subseteq \Bbb R$. Let $\{I_n\}_{n\in\Bbb N}$ be a sequence of disjoint open intervals. I want to prove that $\left|\bigcup_{n=1}^\infty ...
-1
votes
0answers
19 views

The proposition which characterizes a subset of $\mathbb{R^n}$ using a closed set in measure theory.

This is one of the important propositions in the Lebesgue measure theory, Proposition Let $E$ be a subset of $\mathbb{R^n}$. Then, for all $\epsilon >0,$ there exists an open set $G$ such that $E\...
2
votes
0answers
28 views

The property of Lebesgue inner measure.

For bounded set $E \subset \mathbb{R^n},$ difine the Lebesgue inner measure $m_*$ as below. $m_*(E):=m^*(Q)-m^*(Q\cap E^c) \big(=v(Q)-m^*(Q\cap E^c)\big)$ where $Q$ is a closed hypercube in $\mathbb{R^...
-2
votes
1answer
160 views
+50

Does my rigorous definiiton make sense and give what I want? How do we simplify my definiton?

My previous measure in this post doesn't make sense so I made modifications. I need someone to check whether my definitions improved. Suppose we have the following definition? Definition $\ell$ is ...
-4
votes
0answers
31 views

Rings of Continuous functions [closed]

In a ring $C(\mathbb R)$, the ideal $O_0$ of all functions that vanish on a neighbourhood of $0$ is a prime ideal?
0
votes
2answers
32 views

Why the Lebesgue outer measure of the boundary of rectangle in $\mathbb{R^n}$ is zero?

Let $A$ be a closed rectangle in $\mathbb{R^n}$ and let $m^*$ be Lebesgue outer measure. And let $\partial A$ be the boundary of $A$. Then, prove that $m^* (\partial A)=0.$ Since $A$ is a closed ...
0
votes
0answers
31 views

Prob. 26, Chap. 2, in Royden's REAL ANALYSIS: Proof of $m^*\left(A\cap\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty m^*\left(A\cap B_k\right)$

Here is Prob. 26, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $\left\{ E_k \right\}_{k=1}^\infty$ be a countable disjoint collection of measurable sets. ...
0
votes
0answers
38 views

Example of strictly additive outer measure

I am having a difficulty in understanding the proof of "strict sub-additivity" of outer measure. While I took the help of internet, here I found the same example that my teacher gave on his ...
0
votes
1answer
55 views

Find a Borel subset $E$ of $[-1,1]$ s.t. $\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\alpha$ — following up a response. [duplicate]

I am trying to understand the following solution to this question. The question was Let $\alpha \in (0,1)$. Find a Borel subset $E$ of $[-1,1]$ s.t. $$\lim_{r\to 0^{+}} \frac{m(E\cap [-r,r])}{2r}=\...
0
votes
0answers
28 views

Prove $\gamma$ is an outer measure.

$S:$ be a metric space. $\pi \subset \mathcal{P} (\mathcal{S}, \mathcal{B} (\mathcal{S}) = \{ \text{ probability measure on } (\mathcal{S}, \mathcal{B} (\mathcal{S}))\}$. If $\pi$ is tight then it is ...
1
vote
1answer
21 views

Question on weak finite additivity of outer measure.

Let $(X, \mathcal A, \mu)$ be a complete measure space and $\mu^*$ be the outer measure induced by $\mu$ i.e. for any $E \subseteq X,$ $$\mu^*(E) : = \inf \left \{\mu (G)\ |\ G \supseteq E,\ G \in \...
2
votes
0answers
38 views

Positive measure of a Lebesgue measurable set

Here E is a Lebesgue measurable set in $R$. Show if the following is true or false: Every uncountable measurable sets must have positive measure. Every set with positive outer measure is Lebesgue ...
2
votes
0answers
53 views

Prob. 19, Chap. 2, in Royden's REAL ANALYSIS: For a nonmeasurable set $E$ of finite outer measure there exists an open set $O \supset E$ such that …

Here is Prob. 19, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $E$ have finite outer measure. Show that if $E$ is not measurable, then there is an open ...
2
votes
2answers
43 views

Let $A \subset \mathbb{R}^n, k> 0$. Define $kA = \{kx : x \in A \}$. Show that $m^\ast(kA)=km^\ast(A).$

Let $A \subset \mathbb{R}^n, k> 0$. Define $kA = \{kx : x \in A \}$. Show that $m^\ast(kA)=km^\ast(A).$ Assuming that $\mathcal{U}$ is a Lebesgue cover for $A$, then I'm first trying to show that $...
1
vote
1answer
46 views

Equivalence of measurable sets

I am reading L.F. Richardson's Measure and Integral. I am unable to see how a corollary of the following theorem follows: Let $\mu$ be a finite, countable additive measure on a field $\mathfrak A\...
2
votes
0answers
36 views

Give an example of failure of the continuity of outer Lebesgue measure

We know that Lebesgue Measure possesses the following continuity property: If ${\{B_k\}_{k=1}^{\infty}}$ is a descending collection of measurable sets and $m(B_1)<\infty$ , then \begin{equation} m\...
1
vote
1answer
78 views

Hahn Decomposition Theorem and Jordan Decomposition for Finite Signed Pre-measures

Is there a Hahn Decomposition Theorem and Jordan Decomposition for Finite Signed Pre-measures? Let $A$ be an algebra of sets. A set function $r \colon A \to [0,\infty)$ is called a pre-measure if $r(\...
4
votes
1answer
273 views

Bartle exercise 9.N from elements of integration

The exercise states: Let $X$ be a set, $\mathbf{A}$ an algebra of subsets of $X$, and $\mu$ a measure on $\mathbf{A}$. If $B\subset X$ is arbitrary, let $$\mu'(B)=\inf\{\mu(A):B\subset A\in\mathbf{A}\}...
0
votes
0answers
34 views

Intuition about non measurable sets in $\mathbb R^d$ [duplicate]

I am studying elementary measure theory from the book Real Analysis by Stein & Shakarchi. The book defines $m_{*}(E)$ as the outer measure of some set $E \subset \mathbb{R}^d$ and presents the ...
1
vote
1answer
28 views

characterization of positive measure

Suppose $\nu$ is a real measure on $(X,\mathcal{S})$. Define $\mu:\mathcal{S} \rightarrow [0,\infty)$ by $\mu(E)=|\nu(E)|$ (measure of $\nu(E)$ ) We need to show that $\nu$ is positive measure on $(X,\...
1
vote
1answer
49 views

Show that there exist $G,H \in \sigma (\mathcal A_0)$ with $H \subseteq A \subseteq G$ such that $\mu^* (G \setminus H) = 0.$

Let $\mathcal A_0$ be an algebra of subsets of a set $X$ and let $\mu$ be a finite measure on $\mathcal A_0.$ Let $\mu^*$ be the outer measure induced by $\mu.$ Let $A \subseteq X$ be such that for ...
1
vote
1answer
47 views

Can we say that $\mathcal A = \mathcal A^*\ $?

Let $\mathcal A_0$ be an algebra of subsets of a set $X$ and let $\mu$ be a probability measure on $\mathcal A.$ Let $\mu^*$ be the outer measure induced by $\mu$ such that for all $A \subseteq X$ we ...
1
vote
0answers
15 views

Proving that $(X, \mathcal A, \mu^*)$ is the completion of $(X, \sigma (\mathcal A_0), \mu^*).$

Let $\mathcal A_0$ be an algebra of subsets of a set $X$ and let $\mu$ be a probability measure on $\mathcal A.$ Let $\mu^*$ be the outer measure induced by $\mu$ such that for all $A \subseteq X$ we ...
0
votes
0answers
22 views

Can we say that $\mu^*$ is continuous from below?

Let $\mathcal A_0$ be an algebra of subsets of $X.$ Let $\mathcal A$ be the countable unions of elements of $\mathcal A_0.$ Let $\mu$ be a probability measure on $\mathcal A.$ For $A \subseteq X$ ...
1
vote
1answer
20 views

Understanding the proof of non-additivity of outer measure

I am readying Axler's Measure, Integration and Real Analysis and I am stuck on a small portion of the following proof which is highlighted in yellow: (2.19) suggests that for any $n$, $\displaystyle \...
0
votes
0answers
26 views

About proof of “Exterior measure of a closed cube is equal to its volume”.

I suppose my question is related to my basic misunderstanding of something. Part of the proof: In the proof we see "Since $Q$ covers itself, $\textbf{we must have } m^*(Q)\leq |Q|$...". ...
0
votes
0answers
10 views

About the volume of a rectangle which is the almost disjoint union of countable rectangles.

Lemma: If a rectangle is the almost disjoint union of finitely many other recrangles, say $R=\cup_{k=1}^{N}R_k$, then $$\lvert R\rvert =\sum_{k=1}^{N}\lvert R_k \rvert.$$ $R$ means the volume. I ...
-1
votes
1answer
21 views

About the theorem for an open subset $O$ of $R^{d}$. [closed]

Theorem: Every open subset $O$ of $R^{d}$, $d\geq1$ can be written as a countable union of almost disjoint closed cubes. Let's take an example for $d=1$. We take the interval $(0,1)$. If I understand ...
1
vote
1answer
39 views

Show that a function $f : X \to [−∞, +∞]$ is measurable with respect to $μ^∗$ if and only if $μ^∗(T)≥μ^∗(T∩{x∈X:f(x)≤a})+μ^∗(T∩{x∈X:f(x)≥b})$

The following is an exercise from Bruckner's Real Analysis: Measurability can be expressed as a separation property. Let $μ^∗$ be an outer measure on a space $X$. Show that a function $f : X \to [−∞,...
0
votes
1answer
31 views

Why should an outer measure give 0 on an empty set?

A standard definition of outer measure on a set $X$ is a function $$\rho:2^X\to[0,\infty]$$ satisfying the following properties: $\rho(\varnothing)=0$ $A\subseteq B\implies\rho(A)\leq\rho(B)$ $\rho(\...
1
vote
1answer
35 views

Where am I wrong about Borel-Cantelli Lemma

I know there are many questions and answers about this lemma. But I can not grasp the idea. Let my countable collection of measurable sets will look like (-1/n,1/n) where n goes from 1 to infinity. ...
1
vote
1answer
56 views

A nonmeasurable subset of a Hamel Basis

The following is an exercise from Bruckner's Real Analysis: Let $H$ be a Hamel basis and $H_0$ a nonempty finite or countable subset of $H$ . Show that the set of rational linear combinations of ...
1
vote
1answer
55 views

Show that $\inf {\{m^∗(A \cap B): A, B \ m^∗ \text{–measurable}, A \supseteq E, B \supseteq X\backslash E}\} > 0$.

The following is an exercise from Bruckner's Real Analysis: Let $m^∗$ be an outer measure on a set $X$ , and suppose that $E ⊂ X$ is not $m^∗$–measurable. Show that $\inf {\{m^∗(A \cap B): A, B \ m^∗ ...
0
votes
1answer
64 views

Show that for any $E=E_1 \cup E_2$, $μ(E)=μ^∗(E_1)=μ^∗(E_2)$

The following is an exercise from Bruckner's Real Analysis: Let $E$ be a measurable set of positive Lebesgue measure. Show that E can be written as the disjoint union of two sets $E = E_1 ∪ E_2$ so ...
2
votes
1answer
94 views

An example for $m^{*}(\bigcap_{k=1}^{\infty} A_k) \ne \lim_{k \to \infty} m^{*}(A_k)$? [duplicate]

It's a theorem that Let $(X, M, m)$ be a measure space and let $\{A_k\}$ be a sequence of measurable sets. If $A_1 \supset A_2 \supset\dots$ and $m(A_N)<\infty$ for some $N \in \mathbb{N}$ then $m(...
1
vote
1answer
77 views

About some proofs using epsilon notation.

There are many proofs in analysis where we find something like: "M=epsilon. This equality holds for any epsilon>0. So M=0.". What is the background of this conclusion? Maybe I took the ...
1
vote
1answer
46 views

Regular outer measure resricted to a measurable subset

I was wondering if a regular outer measure remains a regular outer measure when the space is a measurable subset of the original space. Rigorously speaking : Let $X$ be our embedding space and let $μ^∗...
0
votes
1answer
26 views

$X$ is locally compact, $\sigma$-compact, $T_2$. $\mu$ be a borel measure which is outer and finite on compact sets. Then $\mu$ is inner measure.

$X$ is $\sigma$-compact then let $X=\cup K_n$ where $K_n$'s are compact. We denote borel $\sigma$-algebra on $X$ by $\mathfrak{m}$. Let $E\in\mathfrak{m}$. We have to show that $$\mu(E)=\text{sup}\{\...
0
votes
0answers
22 views

$A\subset (b,c)$ and $A$ is Lebesgue measurable. prove that $m^*(A)+m^*((b,c)\cap A^c)=c-b$. here $m^*$ is the outer measure.

Let $(X,S)$ be a measure space. $A\subset (b,c)$ and $A$ is Lebesgue measurable. prove that $m^*(A)+m^*((b,c)\cap A^c)=c-b$. here $m^*$ is the outer measure. my work- since $A$ is Lebesgue measurable ...
1
vote
0answers
51 views

Prob. 22, Chap. 2, in Royden's REAL ANALYSIS: $m^{**}(A) = m^*(A)$

Here is Prob. 22, Chap. 2, in the book Real Analysis by H. L. Royden & P. M. Fitzpatrick, 4th edition: For any set $A$, define $m^{**}(A) \in [0, \infty]$ by $$ m^{**}(A) = \inf \left\{ m^*(O) \, ...
1
vote
0answers
50 views

Prob. 20, Chap. 2, in Royden's REAL ANALYSIS: Does $E$ necessarily have to have finite outer measure?

Here is Prob. 20, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $E$ have finite outer measure. Show that $E$ is measurable if and only if for each open ...
4
votes
2answers
142 views

Prob. 18, Chap. 2, in Royden's REAL ANALYSIS: If $E$ has finite outer measure, then there is an $F_\sigma$-set $F$ and a $G_\delta$-set $G$ with …

Here is Prob. 18, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $E$ have finite outer measure. Show that there is an $F_\sigma$ set $F$ and a $G_\delta$ ...
1
vote
0answers
36 views

$A$ is Lebesgue measurable iff $\forall \epsilon > 0$, there exists open set $G$ such that $|G \setminus A| + |A \setminus G| < \epsilon$

In Sheldon Axler's Measure Theory book, I came across this problem - Suppose $A \subset \mathbb{R}$ and $|A| < \infty$. Prove that $A$ is Lebesgue measurable if and only if for every $\epsilon >...
1
vote
1answer
34 views

How to prove this set as a null set?

Suppose $E \subset \mathbb{R}^1$, and $\exists q: 0<q<1 $, s.t. for every interval $(a, b)$, there is a sequence of open intervals $\{I_n, n \ge 1 \}$: $E \cap (a, b) = \cup_{n=1}^{\infty} I_n, \...
1
vote
0answers
33 views

$A,\ B\subseteq R$ and $m*(A)<\infty$, show that $m^*(B\cap A^c)\geq m^*(B)-m^*(A)$

1.$A,\ B\subseteq R$ and $m^*(A)<\infty$, show that $ m^*(B\cap A^c)\geq m^*(B)-m^*(A)$; here $m^*$ is the outer measure. My work- we can write, $m^*(B)\leq m^*((B\cap A^c)\cup A)\leq m^*(B\cap A^...
0
votes
0answers
21 views

Show that $m^*[(a,b)]=m^*[[a,b)]=m^*[(a,b]]=b-a$ here $m^*$ is the outer measure.

Show that $m^*[(a,b)]=m^*[[a,b)]=m^*[(a,b]]=b-a$ here $m^*$ is the outer measure. $My\ work\ - method\ 1$ $(a,b)\subset (a-\epsilon,b+\epsilon)\cup\emptyset\cup\emptyset...\ be \ an \ open\ intervals\ ...
2
votes
1answer
57 views

$A,B\subset\mathbb{R}, |A|<\infty\Rightarrow |B-A|\geq |B|-|A|,\ |\cdot|$ outer measure

I have proved the following statement and I would like to know if I have made any mistakes, thanks. "$A,B\subset\mathbb{R}, |A|<\infty\Rightarrow |B-A|\geq |B|-|A|$" where $|\cdot|$ ...
1
vote
2answers
38 views

Range of measure function cannot be half open interval

Explain why there does not exist a measure space $(X,S,\mu)$ with the property that $\{ \mu(E):E \in S \}= [0, 1)$ My approach was the following - For every $n \in \mathbb{N}$, there is a set $E_n \...
2
votes
0answers
40 views

Prob. 8, Chap. 2, in Royden's REAL ANALYSIS: The collection of finitely many open intervals covering the rational numbers in $[0, 1]$ …

Here is Prob. 8, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $B$ be the set of rational numbers in the interval $[0, 1]$, and let $\left\{ I_k \right\}_{...

1
2 3 4 5
9