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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59 views

Meaning of $\int_{0}^{1} f(x) \mathrm{d}x $ and $\int_{0}^{1} \mathrm{d}f(x) $

Does $\int_{0}^{1} f(x) \mathrm{d}x $ and $\int_{0}^{1} \mathrm{d}f(x) $ mean the same thing? Does $\int_{0}^{1} \mathrm{d}f(x) $ has any meaning unless $f$ is a measure? Any help would be appreciated....
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Outer Measure Generated by Dirac Measure

This is exercise $1$ from chapter 5 in Bauer's book: Let $\mu=\varepsilon_{\omega}$ be the dirac premeasure on a ring $\mathcal{R}$ in $\Omega$. Suppose there exist a sequence $(B_n)\subset \mathcal{R}...
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Finding dominating function

Let $f_n:(0,1)\rightarrow\mathbb{R}$ be $n$ for $x\in(0,1/n)$ and be $0$ otherwise. Then why this $f_n$ have no dominating function which is lebesgue integrable? I want that dominating function ...
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The outer measure of 2 disjoint bounded sets

When $d(A,B)>0$, we have $$m^*(A)+m^*(B)=m^*(A\cup B)$$I want to examine whether the identity holds if we only have $A\cap B=\varnothing$. My attempt: The proof of the origin theorem requires a $\...
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Question regarding to a step of a proof of implying countable cases from finite cases

I have a question regarding to a specific step of the following proof. I am confused to how did $\sum$ for all n implies $\sum $ for countable cases? My initial thought is math induction. However, ...
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“Lebesgue Probability Measure” (Condensed Version) [closed]

Introduction Consider $f:A\to\mathbb{R}$ where $A\subseteq[a,b]$, $a,b \in \mathbb{R}$ and $S\subseteq A$. Here $S$ is a fixed subset of $A$ There is one part of the Lebesgue Measure and Integral I ...
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Purpose of extension of measures?

I have recently learned some abstract concept of induced measures such as outer measure, inner measure, outer measurable, and complete measures on sigma-ring. I am confused as to what is the purpose ...
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Exists an outer measure extenstion $\alpha^{*}$, i.e $\left.\alpha^{*}\right|_{\Xi} = \alpha \iff \alpha$ is $\sigma$-subadditive on $\Xi$

Since I don't wether the definition I have are the standard ones, we define An outer measure $\mu : \mathcal{P}(X) \longmapsto [0,+\infty]$ every $\sigma$-subadditive map, where given $\Xi \subset \...
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for all $E \subset X$ and $\epsilon>0$ there exists $A$ such that $u^*(A) < u^*(E) + \epsilon$ where $u^*$ is an outermeasure

The full problem statement is as follows: Let $\mathcal{A}$ be an algebra on X. let let $\mathcal{A}_{\sigma}$ be the set of countable unions in $\mathcal{A}$. Let $u_{0}$ be a premeasure on $\mathcal{...
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Given any $\varepsilon > 0$, it is possible to cover $\textbf{Q}$ by a countable number of intervals whose total length is less than $\varepsilon$.

The book which I am reading says One consequence of the fact that $m^{*}(\textbf{Q}) = 0$ is that given any $\varepsilon > 0$, it is possible to cover the rationals $\textbf{Q}$ by a countable ...
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Isn't the Lebesgue measure space complete?

Let $X$ be a non-empty set. Let $\mathcal A$ be an algebra of subsets of $X$ and $\mathcal S (\mathcal A)$ be the $\sigma$-algebra of subsets of $X$ generated by $\mathcal A.$ Let $\mu : \mathcal A \...
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39 views

Proof of dominated convergence theorem

I was going through the proof of the Dominated Convergence Theorem. Now if we have that ($f$$_n$) is a sequence of measurable functions such that $\lvert f_n\rvert$ $\le$ $g$ for all n where g is ...
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$A\subseteq \mathbb{R}$ s.t. $|G\smallsetminus A| = \infty$ for every open cover.

I'm working through Axler's Measure and Integration and in exercise 2D number 3 he asks for an example $A\subseteq\mathbb{R}$ such that for every open set $A\subseteq G$ we have $|G\smallsetminus A| = ...
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Any set $E$ of outer measure zero ($m^{*}(E) = 0$) is measurable.

Any set $E$ of outer measure zero ($m^{*}(E) = 0$) is measurable. My solution Since $A\cap E\subseteq E$ and the outer measure is monotonic, we have that $0\leq m^{*}(A\cap E) \leq m^{*}(E) = 0$. ...
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Application of Caratheodory’s theorem on outer measure

I have been going through Caratheodory’s theorem and its applications in measure theory and stumbled upon the following theorem, Let $\mathscr{C}$ be a finitely additive class and $\lambda$ be the ...
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Asking for a hint: proving a set is measurable

Suppose $b < c$ and $A \subset (b,c)$. Prove that $A$ is Lebesgue measurable if and only if $|A| + |(b,c) \setminus A| = c- b$. where $|A|$ denotes the outer measure of $A$. I know how to prove ...
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Question on a proof regarding the additivity of outer measure

The proof is here. The questions asks us to prove If $ E=\cup I_n$ is a countable union of pairwise disjoint intervals ,prove that $m^*(E)=\sum _{n=1}^\infty l(I_n)$. and the interested direction is ...
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Subset of a measurable set with outer and inner measures

Let $\textbf{A}$ an algebra, $\mu:\textbf{A}\rightarrow[0,+\infty]$ a premeasure and $\mu^*:P(X)\rightarrow[0,+\infty]$ an induced outer measure. Let $A\in \textbf{A}^*$ with $\bar{\mu}(A)<\infty$ ...
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metric outer measure proof diam

Let $(X,d)$ be a metric space, $\mathcal{A} \subset \mathcal{P}(X)$ an algebra and $\mu : \mathcal{A} \to [0,\infty)$ an additive function. For $\delta>0$ view the outer measure $\mu_\delta^*$ on $\...
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Outer measure with symmetric difference

Let $A$ an algebra, $\mu:{A}\rightarrow[0,+\infty]$ a premeasure and $\mu^*:P(X)\rightarrow[0,+\infty]$ an induced outer measure. Prove: Suppose that $E \in X $ such that for every $\epsilon>0$, ...
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Limit exists in extended real line of monotone functions

How do I prove the following statement? I'm reading a book in Real Analysis and one of the proofs given is complete if I have the following: Let $F : \mathbb{R} \to \mathbb{R}$ be increasing and ...
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Formal proof: induced outer measure defined via infimum of measure.

Let $(\Omega, \mathscr{S}, \mu)$ be a measure space and $\nu$ the outer measure induced by $\mu$. Then for any $A \subseteq \Omega$ we have $$\nu(A)=\inf(\mu(B):B\in\mathscr{S}, A\subseteq B)$$ I just ...
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Fremlin Question 132Xh

This is question 132Xh from Fremlin Volume 1: Let $A\subset\mathbb{R}^k$ be non-measurable set with respect to Lebesgue measure $\mu$. Show that there is a bounded measurable set $E$ such that $\mu^{*...
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Prove that $\mu^* (F \setminus E) = 0.$

Let $\mathcal A$ be an algebra of subsets of a set $X$ and $\mathcal S (\mathcal A)$ be the $\sigma$-algebra of subsets of $X$ generated by $\mathcal A.$ Let $\mu : \mathcal A \longrightarrow [0,+ \...
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Supremum of Outer Measure sequence

Let $\mu_n^*$ be a sequence of outer measures $\mu_n^* : P(X) \to[0,\infty].$ Prove that $\mu^*=\sup_n \mu_n^*$ is a outer measure too. If $\mu_n^*$ is a outer measure, then I can conclude that for ...
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Show $\mu^*$ is a measure, and $\int_Xg\circ f d\mu = \int_Y g d\mu^*$

Suppose $(X,\mathcal{M})$ and $(Y,\mathcal{N})$ are two measure spaces and $f:X\rightarrow Y$ is such that $f^{-1}(B)\in\mathcal{M}$ for all $B\in\mathcal{N}$. Suppose $\mu$ is a positive measure on $(...
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Proving the existence of a subset, given the Lebesgue outer measure

If c ∈ R, E is a bounded subset of ℝ such that outer Lebesgue measure m*(E) > 0 and 0 < c < m∗(E). How can I show that there exists a subset F of E such that m*(F) = c?
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I want to prove or disprove “∀ open interval I, m∗(I) = m∗(I∩E)+m∗(I∩E^c)” implies “E is lebesgue measurable”.

Let $E \subseteq\mathbb{R}$. Then For all open interval $I \subseteq \mathbb{R}$ , $m^∗(I) = m^∗(I∩E)+m^∗(I∩E^c) \Rightarrow E \textrm{ is lebesgue measurable}$ I want to solve this statement. ...
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In the Littlewood’s 1st Principle, why do we need the finiteness assumption m∗ (E) < ∞?

Theorem (Littlewood's 1st Principle) Every measurable set of finite measure is nearly a finite union of disjoint open intervals, in the sense (i) If E is measurable and m∗(E) < ∞, then ...
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Show that $\mu^*(E\Delta A)<\epsilon$ then $E\in\mathcal{A}^*$

Let $\mu:\mathcal{A}\rightarrow \bar{\mathbb{R}}$ be a premeasure and let $\mu^*:P(X)\rightarrow \bar{\mathbb{R}}$ be the outer measure generated by $\mu$. I need to show that: $(\forall\epsilon>0:...
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56 views

How to show that $\mathcal S^* \subseteq \mathcal S (\mathcal A) \cup \mathcal N$?

Let $\mathcal A$ be an algebra of subsets of a set $X$ and $\mu : \mathcal A \longrightarrow [0,\infty]$ be a measure on $\mathcal A.$ Let $\mu^*$ be the outer measure induced by $\mu$ and $\mathcal S^...
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Two identities on set theory

I was reading extension of measure from the book 'An Introduction to Measure and Integration' by I.K.Rana. In a theorem of that book he proves how to get a complete measure space from an ordinary ...
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How Lebesgue sigma algebra is different from outer regularity.

Give an example of a non-Lebesgue measurable set $A$, such that there exist an open set $O$, $A\subseteq O$ and $m_*(O)\leq m*(A)+\epsilon $ but $m_*(O\backslash A)>\epsilon $? This question ...
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Can we say that $\mu^* (A) = \mu (A),$ for all $A \in \mathcal A$?

Let $\mathcal A$ be an algebra of subsets of a set $X$ and $\mu : \mathcal A \longrightarrow [0,+\infty]$ be a measure on $\mathcal A.$ Define $\mu^* : \mathcal P(X) \longrightarrow [0,+\infty]$ by $$\...
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Prove discrete measure is an outer measure

Goal: Prove the discrete measure: $$ \mu(A) = \begin{cases} 1, & A \neq \varnothing\\ 0, & A = \varnothing \end{cases} $$ is an outer measure. Little description: I don't ...
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Difference between probability measure and outer measure

Im a little confused with regards to the difference between a probability measure and the outer measure. So firstly to set the scene, given $\Omega = \{ 0,1\}$ we have $\mathcal{F} = \{\emptyset, \{...
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show that $\mu$ is countably monotone if and only if $\mu^*$ is an extension of $\mu$

Let $\mathbf{S}$ be a collection of subsets of a set $\mathbf{X}$ and $\mu : \mathbf{S} \to [0,\infty]$ is a set function. Show that $\mu$ is countably monotone if and only if $\mu^*$ is an ...
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Subadditivity of Lebesgue inner measure

Im in a trouble. I want to prove this proposition: Let $A,B \subseteq \Re^n $ such that $d(A,B)>0$. Then, $ m_*(A\cup B) \le m_*(A) + m_*(B) $ . Where $m_*$ is the Lebesgue inner measure. I don'...
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outer measure and Cartesian product [duplicate]

Let $A$ be a subset of $\mathbb{R}^n$, and let $B$ be a subset of $\mathbb{R}^m$. Note that the Cartesian product $\{(a, b) : a \in A, b\in B\}$ is then a subset of $\mathbb{R}^{n+m}$. Show that $m^{*}...
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Outer measure and g_delta set

I think that my logic is not wrong, but I can't prove this problem completely. How can I correct my solution if my logic is wrong? If there is nothing wrong, how can I complete proof? Oh, Korean word ...
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Outer measure one and non-empty intersection with closed sets

Let $X$ be a compact probability space whose Borel sets are measurable. I want to show that a subset $A$ of $X$ is of outer measure one if and only if $A$ has a non-empty intersection with every ...
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Does the continuity from below hold for inner measure?

In class, I have learned that Lebesgue outer measure has upward measure continuity. That is, if $A_n$ increase to $A$, $m^*(A_n)$ converges to $m^*(A)$ even though $A_n$ is not measurable. I am ...
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A simple property of outer measure

I'm struggle with proof about statement as follow. I tried to prove. But I think that it doesn't work.. PROPERTY: Assume that $m^{*}(A)<\infty$. Then, $\forall \varepsilon>0, \exists \mathcal{...
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An example of a measure and of a set s.t. its measure cannot be approximated by measures of its open subsets

Let $(X, \tau)$ be a Hausdorff topological space and let $\mu$ be a measure on the $\sigma$-algebra of Borel subsets of $X$. The measure $\mu$ is (usually) defined to be outer regular if measure of ...
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Measurable functions on metric spaces and sets with positive inner measure

Let $(\mathbb{X},\mathcal{X})$ and $(\mathbb{Y},\mathcal{Y})$ be metric spaces endowed with their Borel $\sigma$-algebras. Let $\mu$ be a measure on $(\mathbb{X},\mathcal{X})$ and let $f:(\mathbb{X},\...
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Theorem 9.13 An introduction to mathematical analysis

Someone please explain me that "since $c_k$ is arbitary ,we can conclude that.." I mean how we got $\sum_{k=1}^{n} \nu(A_k)\leq \nu(\cup_{k=1}^{n}A_k)$? Thanks in advance!
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If $m^*(E)=0$ Then $E$ Is Measurable

Prove: If $m^*(E)=0$ Then $E$ Is Measurable If $A\subset E$ then $A\cap E\subseteq E$ and by the monotonicity of the outer measure $$m^*(A\cap E)\leq m^*(E)=0$$ On the other hand $A\cap E^C\...
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Why is the following topology of probability measures Hausdorff?

Let $X$ be a Hausdorff topological space. Let $PX$ be the set of all Borel probability measures on $X$. Bogachev's Measure Theory (vol. II, section 8.10.iv) defines the $A$-topology on $PX$ to be the ...
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$E$ measurable and $A \subset E$ implies $m(E) = m_*(A) + m^*(E - A)$

I have the following quesiton: If $E$ is measurable and $A$ is any subset of $E$, then $m(E) = m_*(A) + m^*(E - A)$ where $m^*$ and $m_*$ denote the outer an inner measures, respectively. I know ...
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Exercise 1.3.9 in Donald Cohn's Measure Theory [duplicate]

Let $\pi:\Bbb{R}^2 \to \Bbb{R}$ the projection $\pi(x,y)=x$ and $\mu: P(\Bbb{R}^2) \to [0,+\infty]$ the set function $\mu(A)=m(\pi(A))$ where $m$ is the Lebesgue measure on the real line. Then a set $...

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