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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Prove that there exists a bounded set $A\subset R$ such that $|F|\leq |A| − 1$for every closed set $F\subset A$. [duplicate]

Prove that there exists a bounded set $A\subset R$ such that $|F|\leq |A| − 1$ for every closed set $F\subset A$. $|·|$ is outer measure. Below is my idea. Let $A$ be $[0,1]-\mathbb{Q}$, so $|A|=1$. ...
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Equivalence between Borel regular outer measure and regular measures

I am self learning some measure theory. Some sources like Evan's "Measure Theory and Fine Properties of Functions" define a Borel regular measure on a topological space $X$ as an outer ...
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6 votes
4 answers
152 views

Prove that $\mu_*\leq\mu^*$ where the definition of an inner and outer measure is induced by a measure

I know that similar question has been asked and answered here, here, and here. But I am looking for a different proof based on different definitions. We have the following definition of an outer and ...
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Composition of Lebesgue measurable function and Invertible linear transformation is Lebesgue measurable

Let $f:\mathbb{R}^n\to \mathbb{R} $ is a Lebesgue measurable function and $T\in Gl(n,\mathbb{R})$ then show that $f\circ T$ is also Lebesgue measurable . For Borel measurable functions it's easy to ...
jay sri krishna's user avatar
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Question About The Remark after Proposition 1.4.11 from Measure Theory by Donold Cohn

My Question Define subsets $G$, $G_0$, and $G_1$ of $\mathbb{R}$ by \begin{align*} G &= \{x:x=r+n\sqrt{2}\ \text{for some $r$ in $\mathbb{Q}$ and $n$ in $\mathbb{Z}$}\},\\ G_0 &= \{x:...
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Property of Outer Measure on $\mathbb{R}$

Question From - Axler Measure Theory - Problem 3 - Section 2A Throughout: For $A \subset \mathbb{R},$ $|A|$ denotes the outer measure of $A$ and is defined $|A|=inf\\{\sum_{k=1}^{\infty}\ell(I_k): I_1,...
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In the definition of outer measure, can we replace "open intervals" by "disjoint open intervals"

The definition of the outer measure of a set $A\subseteq\mathbb{R}$ is as follows: $$ |A| = \inf \left\{ \Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k): I_1, I_2,\dots\text{ are open intervals such that }A\...
Tran Khanh's user avatar
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Invariance in outer measures

Recently I was trying to solve this problem. Suppose that $X$ is a set and $\mu^{\star}$ is an outer measure on $2^X$. Let $A \subseteq X$ be a set such that $\mu^{\star}(A)<+\infty$ and suppose ...
Tiago Verissimo's user avatar
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Measurability of the set of elements who belong to a infinite amount of subsets in a sequence [closed]

I've been struggling to prove the following statement: Let (X,$\mathcal{M}$,$\mu$) a finite measure space and let $(A_n)_{n\in\mathbb{N}}$ a sequence of measurable sets in X. Now consider $M$ the set ...
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Is there not translation invariant "measure"

I was reading Sheldon Axler's Real Analysis book and he mentions the following (as does the course I am taking) : There does not exists a function $\mu : \mathcal{P}(\mathbb{R}) \rightarrow [0,\infty]$...
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Prove $\mu$ is an outer measure, find the collection of all $\mu$-measurable sets and the necessary and sufficient conditions s.t. $\mu$ is a measure

I am trying to solve the exercise below. I have managed to prove that $\mu$ is an outer measure following the definition of outer measures ($\mu(\emptyset)=0$, monotonicity, $\sigma$-subadditivity). ...
wilma72's user avatar
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Explanation related outer measure and measurable cover

Theorem C is from the book Measure Theory by Halmos...from Chapter 3. My question: how Sigma finite of E is used in the proof? It will be very much helpful kindly give some hint or explanation.
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Inner and outer measure (Lebesgue Measure in the real line) [closed]

If $E$ is an measureble set in the real line , with $m(E)=\infty$, then $\forall A \subset E$ is true that: $$ m_*(A)+m^*(E-A) = \infty \quad? $$ I think this statement is false!!! Mi attempt: Let $...
Maurício Almeida's user avatar
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Can I decompose a set of positive Lebesgue measure into two subsets which have NO positive measure subsets themselves?

Stated in a non-oneliner way: Let $X\subset\mathbb{R}$ be Lebesgue-measurable with positive measure. Is it possible to find sets $A, B$ with the following properties? $A\cap B = \emptyset$ $A\cup B = ...
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Two properties of continuity of measures

I'm studying from Ambrosio-Tilli "Topics on analysis metric spaces" and i think there is an errata corrige. Can someone help me to find the errata corrige book online if there is? However, ...
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3 votes
2 answers
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Proving a Proposition about outer measures

I want to prove a proposition, that tells us a way to generate outer measures. I will first start with the definition of Definition (outer measure) Let $X$ be a set. An outer measure is a function $\...
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Is there a non-measurable set that you can add to a sigma-algebra and nevertheless remain consistent?

Suppose we have a $\gamma : 2^X \to \mathbb{R}_{\geqslant 0}$ that is monotonic ($A \subseteq B \implies \gamma(A) \leqslant \gamma(B)$), semi-$\sigma$-additive ($A \subseteq \bigcup_{i \in \mathbb{N}}...
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What do this three dots (...) mean in PMA Rudin?

I'm studying analysis with PMA by myself and I encountered a proof that I can't understand. In Principles of Mathematical Analysis written by Walter Rudin (page 307), Can you explain what do the ...
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Are there measures that can be obtain in a similar way to Lebesgue measure? [duplicate]

For constructing Lebesgue measure, first we definite the Lebesgue outer measure and we use Caratheodory's theorem for restricting this outer measure to a measure. Are there more non-trivial measure ...
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Show finite additivity of restricted outer measure

Let be $\mathcal{P}(M)$ the power set of $M$, $\mathcal{R}$ a ring of sets based on subsets of $M$ and $\mu:\mathcal{R}\to[0,\infty]$ a content. For an arbitrary but fixed $A\in\mathcal{P}(M)$ we ...
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Construction of outer measure where the closed unit interval is not measurable and outer measure there is not equal to set function

Occasioned by some exercises I saw about creating outer measures in which [0,1] is not measurable because of the definition of the set function that generates the outer measure, I was wondering if ...
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Is this outer measure the same as the absolutely continuous part? (related to Lebesgue differentiation on more general spaces)

In this paper about Lebesgue differentiation: https://arxiv.org/pdf/1802.02069.pdf they define an outer measure $\mu_{\lambda}(A) = \inf \{ \mu(E): E \in \mathcal{B}(X) \text{ and } \lambda(A \...
gordta_chichrron's user avatar
1 vote
1 answer
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Properties of outer measure for measureable sets

I am working on the following task: Let $\phi:\mathcal{P}(X)\rightarrow[0,\infty]$ be an outer measure on $X$. Let $A,B\subseteq X$ and $A$ or $B$ $\phi$ measureable. Show the following statements: (i)...
Lukas Kretschmann's user avatar
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Is $\mu: \mathcal{P}(\mathbb{N}) \rightarrow [0, \infty]$ a measure or outer measure?

Consider the function $\mu: \mathcal{P}(\mathbb{N}) \rightarrow [0, \infty]$ defined by $$\mu(A) := \limsup\limits_{n \rightarrow \infty} \frac{1}{n} \cdot \#(A \cap \{1, \dots, n\})$$ Is this ...
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Hann decomposition problem, Let $\mu$ a finite measure, such that $\lambda \ll \mu$ and let $P_n, N_n$ Hann decomposition for $\lambda - n\mu$.

I'm trying to solve this Hann decomposition problem, it is as follows: Let $\mu$ a finite measure,such that $\lambda \ll \mu$ and let $P_n, N_n$ Hann decomposition for $\lambda - n\mu$. Let $P = \...
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Showing the union of two measurable sets is measurable with the $G_\delta$ definition

In Chapter $2$ number $21$ of Royden, we are asked to prove the measurability of the union of two measurable sets assuming that a set $E$ is measurable if and only if there exists a $G_\delta$ set $G$ ...
Bifton Mifts's user avatar
1 vote
1 answer
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A descending sequence with a non-additive measure where the measure of the limit isn't equal to the measure of the intersection

Let $\mu^*$ be an outer measure over $\mathbb{R}$ so that $$\mu^*(A) := \begin{cases} 0 & \text{$A$ is finite or countable} \\ 1 & \text{otherwise} \end{cases}$$ A sequence of sets $(A_n)$ ...
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Equivalence of outer lebesgue measure if covering of cubes is required rather than cuboids

I want to prove that the outer lebesgue measure does not change when requiring to cover every set by cubes rather than cuboids. (i.e. See definition here: https://en.wikipedia.org/wiki/...
MathMaestro's user avatar
1 vote
1 answer
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$\tau$ on $X$ with $\tau \leq \rho$ on $\mathcal{E} \implies \tau \leq \mu^*$.

We are supposed to solve the following problem. I think I managed to solve the first part, but I'm struggling with the second one. Let $X$ be a set and $\mathcal{E} \subset \mathcal{P}(X)$ with $\...
Minerva's user avatar
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2 votes
0 answers
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In Lebesgue's characterization of measurability, why is the set required to have finite outer measure?

In Chapter $2$ of Royden, #$20$, we are asked to show for a set $E$ of finite outer measure that $E$ is measurable if and only if for each open bounded set $(a,b)$ we have $m^*((a,b))=b-a=m^*((a,b)\...
Bifton Mifts's user avatar
2 votes
2 answers
323 views

What does "take over" mean in the "the inf being taken over all countable coverings of E by open elementary sets"?

I'm studying real analysis with Rudin Principles of Mathematical Analysis textbook. I'm confused about the expression "the inf/sup being taken over ~~" in several definitions. In principles ...
jjw's user avatar
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1 answer
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Questions about details in Lebesgue Outer Measure Theorem proof for an interval I , $|I|_e=v(I) $

Definition and notation: Consider closed $n$-dimensional intervals $I=\{x:a_j \leq x_j \leq b_j, j=1,2,3,...,n\}$ and their volumes $v(I) =\prod_{j-1}^{n}(b_j - a_j)$.We define the outer measure of an ...
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0 answers
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When a set function on a product sigma-algebra is a measure

Let $(X,\mathcal{A})$ and $(X,\mathcal{B})$ be measurable spaces. Every measure $\mu$ on $(X\times Y, \mathcal{A}\otimes\mathcal{B})$ gives an assignment $\mathcal{A}\times\mathcal{B}\to\Bbb{R}$ via $(...
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1 answer
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9K Bartle $A\subset B$ with $l^*(B\setminus A)=0$

Let $A$ be a borel measurable set. I need to prove the existence of $B$ in the Borelians such that $A\subset B$ and $l^*(B\setminus A)=0$. Because of the way the outer measure $l^*(A)$ is defined (as ...
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Example of a set mapping that is not an outer measure by violating either monotonicity or subadditivity

Let $X$ be a non-empty set and $\mu:2^X\to[0,\infty]$ be a map. Do you happen to know good examples of such $\mu$s that do not satisfy either the monotonicity or countable subadditivity of a proper ...
Cartesian Bear's user avatar
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Definition of Borel regular outer measure.

I have ran across this in the hypothesis of one certain theorem. Let $X$ be a separable metric space with a Borel regular outer measure $\mu^{*}$ such that $\mu^{*}X = 1 $. I don't understand the ...
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How to prove the following outer measure is trivial? [closed]

Define $\rho:\mathbb{R}\to[0,\infty)$ by $\rho((a,b])=(b-a)^\alpha$ for some $\alpha>1$, need to show the outer measure $\mu^*$ induced by $\rho$ is trivial. I think it is sufficient to show $\mu*([...
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Solving Intervals Problem Without Outer Measure

Can the following theorem on intervals be proved by elementary means, without using the outer measure [1, Chap 2] ? Theorem If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ with $\...
Ross Ure Anderson's user avatar
3 votes
1 answer
62 views

Does Carathéodory's extension of measure space preserve inclusion?

A measure space $(X,\mathcal{S},\mu)$ can be extended to a measure space $(X,\hat{\mathcal{S}},\hat{\mu})$ as follows: $\mu$ is extended to an outer measure $\mu^\ast$ (defined as the infimum of the ...
ashpool's user avatar
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Show that if $f:\mathbb{R}\to\mathbb{R}$ is differentiable with $|f'|\leq 1$ then $\lambda^*(f(I))\leq \lambda^*(I)$ for all open intervals $I$.

I'm studying for my Analysis master's exam and I'm trying this question from a past exam. Here is the question: Let $\lambda^*$ be the Lebesgue outer measure on $\mathbb R$. Suppose that $f:\mathbb R\...
blakedylanmusic's user avatar
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Lebesgue outer measure of product of sets

Let $\lambda^\ast$ be the Lebesgue outer measure on $\mathbb{R}$, and $\lambda^\ast_2$ the Lebesgue outer measure on $\mathbb{R}^2$, defined as the infimum of the sum of the rectangles covering a set. ...
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4 votes
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Is the completion of a measure space the same as the Hahn-Kolmogorov extension?

Let $(X,\mathcal{S},\mu)$ be a measure space. I know two ways of extending it to a complete measure space: Forming the "completion" $(X,\hat{\mathcal{S}},\hat{\mu})$, where $\hat{\mathcal{S}...
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Measure spaces that stabilize with respect to the Carathéodory extension

When learning measure theory, for sure one encounters the Carathéodory extension theorem with extends a premeasure defined over a semiring to a measure defined over a $\sigma-$algebra. Let $X$ be a ...
Jianing Song's user avatar
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7 votes
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Is there a nonmeasurable subset of $\mathbb{R}^2$ that is $1-$dimensional Hausdorff measurable?

For $n\in\mathbb{N}^*$ and $s\in\mathbb{R}_{\ge 0}$, the $s-$dimensional Hausdorff measure $H^s$ is an outer measure over $\mathbb{R}^n$, and the $\sigma-$algebra of $s-$dimensional measurable subsets ...
Jianing Song's user avatar
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1 vote
1 answer
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Can the measure restricted from an outer measure extend to a different outer measure?

Let $X$ be a set, and $\rho$ an outer measure on $X$. Let $\mathcal{M}(\rho)$ be the set of $\rho$-measurable sets. Then the restriction $\rho|_{\mathcal{M}(\rho)}$ is a measure on $\mathcal{M}(\rho)$....
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Extending a finite measure from a semi-ring $\mathcal{E}$ to $\sigma(\mathcal{E})$

Let $\mathcal{E}$ be a semi-ring of subsets of $\Omega$ and let $\mu:\mathcal{E}\rightarrow [0,\infty]$ be a measure on $\mathcal{E}.$ Define the outer measure $\mu^{*}$ generated by the measure $\mu$ ...
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Measure not equal to set function that generates outer-measure Counterexample

Let $ M\subseteq 2^X$ be a collection of subsets of $X$ and $\rho:M\to [0,\infty]$ be a set function such that $\emptyset,X\in M$ and $\rho(\emptyset)=0$. The set function $\rho$ induces an outer ...
ECON10105's user avatar
2 votes
0 answers
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A lemma used to prove Besicovitch Theorem

Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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A question about Theorem 2.8.2 of "Geometric Measure Theory" of Federer.

Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
Grace53's user avatar
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1 vote
1 answer
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A question about Theorem 2.8.7 from "Geometric Measure Theory" of Federer

Let $X$ be a metric space and $\mu\colon\mathcal{P}(X)\to[0,+\infty]$ an outer measure over $X$ such that all open subsets of $X$ are $\mu$-measurable (then $\mu$ is a Borel measure) and $\mu$ is ...
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