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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Volume of an interval in $R^N$ seen as the supremum of set of volumes of compact subintervals

In a Lemma to be used in developing properties of Lebesgue Outer Measure on $R^N$ ($N$ is a positive integer) the professor pointed that it is not hard to establish that the volume of an interval $I$ ...
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Measure Theory question about Outer measure vs measure

We are currently covering the last chapter of baby Rudin where he quickly covers the basics of Measure Theory. I am having a hard time understanding under what circumstances does the outer measure ...
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Are these two families of sets the same without the assumption of measurability?

Suppose we have a nonempty set $X$ and an outer measure $\mu$ on $\mathcal{P}(X)$. Let's denote the $\sigma$-algebra of $\mu$-measurable sets, i.e. $\sigma$-algebra from Caratheodoty extension theorem,...
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An interesting qn came in my mind related to outer measure of the cartesian product of A and B??

So here are the questions came in my mind, but I could not answer each of them.. If $A$ and $B$ are two subsets of $\Bbb{R}$ then $\lambda_2^*(A×B)=\lambda_1^*(A) \lambda_1^*(B)$?? (If not the case, ...
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For any set $E\subset\mathbb{R}^{p+q}$, if for all $x\in\mathbb{R}^p$ that $E(x)$ is measurable, does $E$ measurable?

The definition $E(x)=\left\{y\big|y\in\mathbb{R}^q\land <x,y>\in E\right\}$ is all the points that in E while $x$ is selected. And furthermore, $E$'s projection in $\mathbb{R}^p$ is also an ...
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1.3.9 from the book measure theory of Donald L. Cohn

I've read the solution of this exercize from this link Cohn Measure Theory exercise question 9 chapter 1.3 other direction When I've tried to prove that $\lambda^* ( B_1 - B_0 ) = 0$ with the hint ...
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Show that induced outer measure of the sum of two measures is the sum of the induced outer measures

Let $H$ be a semiring over $\Omega$ and let $\mu$ and $\nu$ be two measures on $H$. Show that $(\mu + \nu)^* = \mu^* + \nu^*$ with the induced outer measures $\mu^*$ and $\nu^*.$ We defined the ...
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Outer measure "distributive" property

Let $X$ be a set, $\mathcal{A}\subseteq 2^X$ is a ring of sets of $X$, $\mu: \mathcal{A}\to [0, \infty]$ is a countably additive function. And let $\mu^*$ is outer measure induced by $\mu$ on $2^X$. ...
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Prove $E_2$ is measurable.

If $E_1$ measurable and $m^* (E_1 \oplus E_2)=0$, then $E_2$ measurable. Note: $E_1 \oplus E_2 = (E_1\cup E_2)-(E_1 \cap E_2)$. $E_1$ measurable if for all $A\subset \mathbb R$ satisfy $m^*(A)=m^*(A\...
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How is a restricted measure defined over sets not in sub-sigma algebra?

According to ProofWiki given a measure space $(\mathsf{X}, \mathcal{X}, \mu)$ and a subsigma algebra $\mathcal{Y}\subseteq\mathcal{X}$, the restriction of $\mu$ to $\mathcal{Y}$ is the measure $\mu\...
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Prove $E$ is lebesgue measurable set and calculate lebesgue measure of E

Can someone help me with the following problem? Let $E$ to be to set of all real numbers in the interval $[0,1]$ which in thier decimal expension the digit $"1"$ appears only in a finite number of ...
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Outer measure does not satisfy countable additivity

Let $\Omega \subseteq \Bbb R^n$, we define the outer measure the following way : $m^*(\Omega):= \inf\{ \sum_{j \in J} \text{vol } B_j \lvert \{B_j\} \text{ is countable covering by boxes of } \Omega \}...
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$E$ measurable implies $\forall \varepsilon>0\exists A\in\mathcal S_\delta$ s.t. $A\subseteq E$ and $\mu^*(E\smallsetminus A)<\varepsilon$.

From Royden's Real Analysis 4th edition, section 17.5, problem 36: Let $\mu$ be a finite premeasure on an algebra $\mathcal S$, and $\mu^*$ the induced outer measure. Show that a subset $E$ of $X$ is $...
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Compact interval $I=[a,b]\subseteq \mathbb{R}$ is an $\mathfrak{L}$-measurable set. [closed]

Let $\mathfrak{L}$ denote the Lebesgue (outer) measure on $\mathbb{R}$. Show, directly using the definition of $\mathfrak{L}$, that a compact interval $I=[a,b]\subseteq \mathbb{R}$ is a $\mathfrak{L}$-...
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Show that every probability on $\mathbb{f(R)}$ for f a subjective continuous function is the pushforward measure of a measure by that function

I really could use a hand on this question Let $f:\mathbb{R} \mapsto f(\mathbb{R}):=E $, a continuous function. Let $\mu_e$ be a probability on $(E,B(E))$. Show that there exists a measure $\nu$ on $\...
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Show that the Hausdorff measure on $R^n$ does or doesn't depend on the norm utilised

I have been having trouble on this question for a while now. We define the Hausdorff on $B_\mathbb{R^n}$ measure this way : $H^\beta(F) = lim_{\delta\to 0} \inf{\{\sum diam(U_i)^\beta : \delta \ge ...
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Understanding when a set is outer-measurable

Suppose $\mu$ is an outer measure on some arbitrary set $X$. We say a subset $E$ of $X$ is $\mu$-measurable if and only if $$\mu(A) = \mu(A \cap E) + \mu(A \setminus E), \quad \forall A\subset X.$$ I ...
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Lebesgue Outer Measure Additivity for disjoint intervals.

I am trying to prove that if $\cup_{i=1}^{\infty}I_i$ are disjoint open intervals then $$|\cup_{i=1}^{\infty}I_i|=\sum_{i=1}^{\infty}\mathcal{l}(I_i).$$ Where $||$ is the Lebesgue outer measure. I ...
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Lebesgue Outer Measure Null Set Proof

I am trying to solve the following proof: Let $||$ be the Lebesgue outer measure. Show that if $|B|=0$ then $|A\cup B|=|A|$. If I consider $B=\{x_1,x_2,....\}$ this is easy enough. Consider an ...
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For bounded $A \subset \mathbb{R}$ we define: $ m_*(A) := m(E)-m^*(E\setminus A), E\subset R,A\subset E, m(E)<\infty$ show that $m_*$ is well defined

I was trying to solve the following question: For bounded $A \subset \mathbb{R}$ we define: $$ m_*(A) := m(E)-m^*(E\setminus A)$$ For Lebesgue measurable set $E \subset \mathbb{R}$, where $A\subset E$ ...
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If $A$ and $G$ are disjoint subsets of $\mathbb{R}$ and $G$ is open, $|A \cup G| = |A| + |G|$

I attached the book link here, you may check the proof on Page 47. Additivity of outer measure if one of the sets is open. Suppose $A$ and $G$ are disjoint subsets of $\mathbb{R}$ and $G$ is open. ...
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Examples of finite outer measures induced by a measure on a ring.

I'm looking for as much examples as possible of the following. Consider a set $X$, a ring of subsets $\mathscr{R}\subseteq X$ and a function $\mu:\mathscr{R}\to\mathbb{R}_{\geq0}$ such that: $\mu(\...
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Does the outer measure of the product equal the product of outer measure?

Assume $A\subset\mathbb{R}^p,B\subset\mathbb{R}^q$, then $A\times B=\left\{(x,y)\big|x\in A\land y\in B\right\}\subset\mathbb{R}^{p+q}$ Then is it true that $m^*(A\times B)=m^*(A)m^*(B)$? Here $$m^*(E)...
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Sets in completed measure space equals additive sets

I‘m trying to prove that the sets in the completed (finite) measure space $(X,\mathcal{E},\mu)$ are the additive sets. Definitions: We have defined a set $A\in\mathcal{P}(X)$ as additive if: $$\mu^{*}(...
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Rearranged sequences to prove countable subadditivity of outer measure

I'm working through the countable subadditivity of outer measure proof in Axler (Measure, Integration & Real Analysis), and I'm finding it difficult to understand the purpose or intuition of the ...
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The Motivation of Pre-measure for Construction of Measures

It seems to me that there at least two different ways to "construct" a measure. The first way is to first define an outer measure using a set function and then "restrict" the outer ...
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Does Sigma Algebra Necessarily Induce a Measure?

I am wondering if we have a space $X$ and an outer measure $\mu^*$ defined on $P(X)$, is it always true that the restriction $\mu^*|_\mathcal{F}$ is in fact a measure for an arbitrary $\sigma$-algebra ...
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If $E\in \mathscr{M}_{\mu^*}$ , then for each $\varepsilon$ exists $A\in \mathscr{A}$ such that $\mu^*(A\triangle E)< \varepsilon$

Let $X$ be a set, $\mathscr{A}$ a ring of subsets of $X$, $\mu ∶ \mathscr{A} \to \overline{\mathbb{R}}_{≥0}$ a premeasure and $\mu^*$ the outer measure generated by $\mu$. (By Caratheodory) If $E\in \...
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Prove $\mu^*(A)=\nu(A)$ if there exists a cover $A\subset \cup_{n\geq1} B_n$ and $\mu^*(B_n)<\infty \;\forall n\geq 1$

Given $\mu : \mathscr{H} \to \mathbb{R}$ a pre-measure on $\mathscr{H}\subset X$ a semiring and $\mu^*$ the outer measure defined by $\mu$, $\mathscr{A}=\sigma(\mathscr{H})$ and $\nu$ a measure such ...
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Extension of measure is less or equal to outer measure when restricted to semiring [closed]

If $\mu : \mathscr{H} \to \mathbb{R}$ is a pre-measure on $\mathscr{H}\subset X$ a semiring and $\mu^*$ the outer measure defined by $\mu$, $\mathscr{A}=\sigma(\mathscr{H})$ and $\nu$ a measure such ...
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Show that $\mu^*(M\cup N)=\mu^*(M)+\mu^*(N)$ for outer measure $\mu^*$

Let $\mu:\mathscr{H}\to\mathbb{R}$ be a content where $\mathscr{H}\subset X$ a semiring and $\mu^*$ the outer measure defined by $\mu$. I have already proven that that for any $A,B\subset X:\quad \mu^*...
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Prove relations for a content $\mu: \mathcal H \to \mathbb R$ where $\mathcal H$ is a seminring over the set $\mathit X$

I have the following three relations to show in my measure theory exercise course. Let $\mu: \mathcal H \to \mathbb R$ be a content on the semiring $\mathcal H$ over the set $\mathit X$ and $\mu^{*}$ ...
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continuity of outer measure

Let $\{E_n\}_{n=1}^{\infty}$ be an increasing sequence of subsets in $\mathbb{R}$ ($\textbf{not}$ necessarily measurable). Show that $ m^*(\bigcup_{n=1}^{\infty}E_n) = \lim_{n \to \infty } m^*(E_n) $ ...
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Why do we need hypothesis of complete measure in this version of Fubini's theorem?

I'm reading below Fubini's theorem in page 3 of this lecture note. Let $(X, \mathcal{A}, \mu)$ and $(Y, \mathcal{B}, \nu)$ be complete measure spaces, let $\gamma$ be the product outer measure on $X \...
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Does this construction of an outer measure extend the original function?

Given a set $X$, a collection of subsets $K\subseteq \wp(X)$ such that $\varnothing\in K$ and a function $m:K\to[0,\infty]$ such that $m(\varnothing)=0$, we define the function $\lambda:\wp(X)\to [0,\...
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Explicitly finding a complete measure space

Let $(r_n)_{n = 1}^{\infty}$ be an enumeration of the rationals. Define $\mu = \sum_{n \ge 1} 2^{-n} \delta_{r_n}$ so that $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ is a measure space, where $\...
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Why is the inner measure smaller than/equal the outer measure? If I try to prove it, I always end up the other way around, no matter how I try

We have the outer measure $$\mu^{*}(A) = \inf \left\{ \sum_{n=1}^{\infty} \mu_0 (A_n) \: | \: A_n \in \mathcal{A}_0, A \subset \bigcup_{n=1}^{\infty} A_n \right\}$$ The inner measure is: $$\mu_{*} (A) ...
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Lebesgue differentiation theorem for monotone functions via Vitali covering lemma

I was reading LECTURES ON LIPSCHITZ ANALYSIS by Juha Heinonen and at the Theorem 3.2 he gives a proof of Lebesgue Differentiation Theorem for monotone functions. He says that we can easily (using the ...
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Measure Theory Specially Carathéodory measure. and the sigma algebra induced by that [closed]

Let $X$ be a set, and let $T$ be a $\sigma$-algebra on $X$. Let $f: T \to [0,1]$ be a measure and let $\mu∗ f$ be the Carathéodory outer measure induced by $f$. Is it always the case that $T = T\mu∗ f$...
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How do I prove this claim about the given measure?

I am working on the following problem: Let $\Omega=[0,1]^2$ and $\mathfrak{R}$ be the ring in $\Omega$ generated by the cuboids $[a,b)\times [0,1]$ with $0\leq a<b\leq 1$. We define a measure $\mu$ ...
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How to determine the Carathéodory outer measure $\mu^*$ induced by $\mu$ and the $\sigma$-algebra of $\mu^*$-measurable sets?

I encountered such an exercise: Let $H=\chi_{[0,+\infty)}$ be the Heaviside function, and $\{r_n\}_{n=1}^\infty$ is a given real sequence. Let $$\mathcal{R}:=\left\{\bigcup_{i=1}^k (a_i,b_i] : k \in ...
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Why we need $A$ to be with finite outer measure? what will be disturbed in the proof if this condition does not occur?

Here is the question I wanna answer the backward direction of it: Let $E$ have finite outer measure. Show that $E$ is measurable if and only if for each open, bounded interval $(a,b),$ we have $$ b - ...
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Why does the vitali set not have 0 outer measure

We know that any set with zero outer measure is measurable, and the vitali set on $[0,1]$ is non-measurable. So it must have a non-zero outer measure. $\textbf{Lemma: }$ For any $q \in \mathbb{Q}$ ...
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2 votes
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Assume that $A\subset\mathbb{R}$, $0<c<m^* (A)$, can we find a subset $B$ of $A$ such that $m^*(B)=c$?

Here $m^*(A)$ denotes the outer measure of $A$. I know the proposition is true for measurable $A$, see If $X \subset \mathbb R$ is measurable then for every $\alpha \in (0, \mu(X))$ there exists $X_\...
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If two sets have the same outer measure, do their images under a function (or linear map) have the same measure?

I'm studying for an exam and thinking about the following question: If $E,E' \in \mathbb{R}^d$ and $m_*(E)=m_*(E')$, is $m_*(f(E))=m_*(f(E'))$, where $f$ is a function or linear map? I don't think it'...
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A confusion regarding the proof of "Lebesgue outer measure of an interval is its length."

I have understood the overall proof but I have a confusion at the very beginning of the proof: Usually in the first case of a closed interval, say $[a,b]$, we find that for each $\varepsilon>0,$ $\...
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Exterior Lebesgue measure example

Let $\lambda^*$ be the exterior Lebesgue measure on $\mathbb R$ and $A\subset \mathbb R$ We define $\lambda^*$ : $$ \large \lambda^* = \text{inf}\Bigg\{\sum_{i=1}^{+\infty}{(b_i-a_i}) |A \subset \...
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Inclusion-Exclusion Principle for Outer Measure

Question: Let $\mu^*$ be an outer measure on a set $\Omega$ and $E$ be a $\mu^*$-measurable set. Show that $$ \mu^*(A) + \mu^*(E) = \mu^*(A \cap E) + \mu^*(A \cup E) $$ for all $A \subseteq \Omega$. ...
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Outer Measure and $\sigma$-algebra on $\Omega = \Big\{(i, j)\ \big|\ 1\le i,j \le N\Big\}$

Question: Let $N\in\mathbb{N}$ and $$ \Omega = \Big\{(i, j)\ \big|\ 1\le i,j \le N\Big\}\subseteq \mathbb{N}\times\mathbb{N} $$ Define $\mu^*: \mathcal{P}(\Omega) \to [0, \infty]$ by letting $\mu^*(A)$...
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Show that the collection of sets for which the inner measure equals the outer measure $\mu_*(A) = \mu^*(A)$ is a $\sigma$-algebra.

On space $\Omega$ we have algebra $\mathcal{A} \subset \mathcal{P}(\Omega)$ with measure $\mu: \mathcal{A} \to [0,1]$ and we define the inner measure $\mu_*: \mathcal{P}(\Omega) \to [0,1]$ and outer ...
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