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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Do we need intervals to define the Lebesgue measure?
The Lebesgue measure is conventionally defined as
$$\mu(X) = \inf\{\sum_{n \in \mathbb{N}}(b_n-a_n) | X \subseteq \bigcup_{n \in \mathbb{N}}(a_n,b_n) \}$$
Which can be thought of intuitively as ...
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Prove that $λ^{∗}_{1} (A) + λ^{∗}_{1} (B) ≤ λ^{∗}_{1}(A + B)$
I have this exercise that I don't know if I solved right.
Let $A, B ⊆ \mathbb{R}$ be Borel-measurable subsets with $\max A = 0 = \min B$. We consider $A + B := \{a + b : a ∈ A \text{and} b ∈ B\}$.
...
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doesn't exist a closed subset $F$ of $[0,1]$ such that $F\subset\mathbb{R}\setminus\mathbb{Q}$, and $|F|=1$ Axler Measure, Integration & Real Analysis
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 13 on p.24 in Exercises 2A in this book.
Suppose $\epsilon>0$. Prove that ...
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Prove that $|(a,b)\cup (c,d)|=(b-a)+(d-c)\text{ if and only if }(a,b)\cap (c,d)=\emptyset.$ ("Measure, Integration & Real Analysis" by Sheldon Axler)
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
The following exercise is Exercise 7 on p.23 in Exercises 2A in this book.
I want to prove this exercise using only ...
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Prove if $A,B\subset\mathbb{R}$ and $|A|<\infty$, then $|B\setminus A|\geq |B|-|A|$ ("Measure, Integration & Real Analysis" by Sheldon Axler)
I am reading "Measure, Integration & Real Analysis" by Sheldon Axler.
2.5 outer measure preserves order on p.16
Suppose $A$ and $B$ are subsets of $\mathbb{R}$ with $A\subset B$. Then $|...
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Are Lebesgue measurable sets the only sets for which the outer measure is sigma additive?
I am currently studying Lebesgue measure theory, and I had some questions about the definition of the Lebesgue measure.
So, to start off, we are looking for a well-defined, positive function $m$ ...
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locally measurable functions
In the books of Boubarki on integration, the autors define $L^\infty$ spaces with the help of locally negligible subsets. In the books of Fremlin (and other classical books), the author does not seem ...
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Outer signed measure
I would like to ask whether there is some kind of analogue of outer measure when dealing with signed measures. I would like to assign measure to all the subsets, not just some $\sigma$-field.
I'm ...
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Density points outside of a non-measurable set [closed]
Let $\lambda$ be a Radon outer measure over $R^n$ (that is: all Borel sets are $\lambda$-measurable in Caratheodory sense, for every set $C$ there exists a Borel set $B$, $C \subset B$ such that $\...
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If $X$ is dicrete and $m$ satisfies probability axioms except $m(\bigcup_{n\in J}A_n)=\sum_{n\in J}m(A_n)$ for finite $J$, does countable sum hold? [duplicate]
Let $X\subset\mathbb{R}^n$ be a non-empty discrete set in the standard topology and $m:\mathcal{P}(X)\to[0,1]$ be a function satisfying all probability measure axioms except that countable summability ...
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Hausdorff Outer Measure on $\mathbf{R}^n$ is a Measure when Restricted to the Lebesgue $\sigma$-Algebra
Here is the problem that I am dealing with specifically:
I was able to prove part (a) and (b), where in part (b), I showed that the Hausdorff Outer measure is a metric outer measure and thus the ...
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Examples of premeasures
I'm looking for a variety of examples of premeasures. I know that length of intervals is one example. I also know that, more generally, we can take any function F which is nondecreasing and so on, ...
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Is the limit superior of Radon measures at least a Radon outer measure?
Let $\left(\mu_n\right)_{n=1}^\infty$ be a sequence of Radon measures on $X\subset \mathbb{R}^n$ such that for every compact subset $K\subset X:\sup_{n=1,2,\dots}\mu_n(K) < \infty$. Is it then true ...
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Intersection with Vitali set is not measurable
So, I have recently stumbled upon the following problem while studying measure theory.
Given a Vitali set $V\subset [0,1]$ and a positive measure set $A \subseteq [0,1]$, prove that their intersection ...
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Are positively separated sets separated?
I was reading this answer where the author does a clever trick to prove that the Lebesgue outer measure is a metric outer measure. I know how to prove that the Lebesgue outer measure is a metric outer ...
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How does one prove outer measure is finite for $A \in \mathfrak{M}_F(\mu)$ if there exist elementary sets $A_n \to A$? [closed]
I was trying to prove for $A \in \mathfrak{M}_F(\mu)$ that if $A_n \to A$ with $A_n$ elementary sets, then $\mu^*(A)$ is finite. (All notations and definitions consistent with Rudin's Principles of ...
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Justifying an inequality in a proof of Carathéodory's Theorem
In the proof below, how is the inequality
$$\lambda(G) \le \sum_n\sum_k\mu_0(F_{n,k})$$
justified?
screenshot, transcribed below:
A1.8. Proof of Carathéodory's Theorem.
Recall that we need to prove ...
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outer measure of product measure is product of outer measures
This question has been asked on this site (e.g. here) in some specific cases, but I would like to check my proof in this more general setting:
Let $(X,\mathcal M, \mu)$ and $(Y,\mathcal N,\nu)$ be $\...
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Caratheodory construction using outer measures - uniqueness
I am currently working on Caratheodory Extension Theorem topic and am confused about the use of uniqueness here. Note the following theorem and corresponding definitions:
For every outer measure $\mu^*...
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Lebesgue outer measure using open balls- help with proof
This question has been asked before but the solutions use a version of Vitali Covering Lemma which is not clear to me how it relates to the statement I am familiar with (below). Any help is ...
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On the abscence of the Inner Measure in introductory texts on Measure Theory
Letting $\mu$ and $\mu^*$ be the Lebesgue and outer Lebesgue measure repsectively, the inner Lebesgue measure can defined as
$$\mu_*:S\mapsto \sup\left\{\mu(K) : \text{$K$ is a compact subset of $S$}\...
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Can Carathéodory's Extension Lemma be proven without use of the Carathéodory's Restriction Lemma?
Either the Extension Lemma or the Restriction Lemma below may be used to directly construct the Lebesgue measure, yet the only proof I know of the former lemma uses the latter. Can the Extension Lemma ...
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Is there a relation between content and outer measure?
Let $S$ be a nonempty set and $R$ a ring over $S$. Furthermore, let $\mu : R → [0, ∞]$ be a content (a measure that is only finitely additive ) on the ring and $\mu'$ be the outer measure Given an ...
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Outer measure of dirac measure
Let $\delta_x$ be the dirac measure and $\delta_x^*$ the corresponding outer measure. Show that for all $A \subset (0,1]$ the equation
$\delta_x^*(A):=\cases{1, x \in A \\ 0, x \notin A}$
holds.
Note ...
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How to prove that $λ^∗_{n+m}(A × B) \leq λ^∗_n (A) λ^∗_m(B)$
I have problems with this exercise. Can someone help me? I really don't know how to start:
Let $A ⊆ R^n$ and $B ⊆ R^m$. Show that if $λ^∗_n (A) \neq\infty$ and $λ^∗_m(B) \neq \infty$, then $λ^∗_{n+m}(...
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Exercise 9.H in Bartle's Elements of integration
If $B$ is a Lebesgue measurable subset of $\mathbb{R}$, if $\varepsilon>0$ and if $B\subseteq I_n=(n,n+1]$, then exists a compact set $K_\varepsilon\subseteq B$ such that
$$l^*(K_\varepsilon)\leq l^...
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$m^\star(A\cup B)<m^\star(A)+m^\star(B)$
Show that there are disjoint sets $A$ and $B$ such that $m^\star(A\cup B)<m^\star(A)+m^\star(B)$
I know how to prove,
If $A$ and $B$ be bounded subsets of $\mathbb{R}$ for which $d(A,B)>0$ ...
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Existence of Outer Measure and its Measurability
It was a big curious for me that whether, for a given set $E\subset\mathbb R^p$ and an outer measure $\mu^*$, the existence of $\mu^*(E)$ implies the measurability of $E$. Plus, the same question if ...
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If $\mu^\ast$ is an outer measure that is finitely additive then $\mu^\ast$ is a measure.
Problem: Let $\mu^\ast$ be an outer measure on $\Omega$ such that for any finitely many disjoint sets $A_1, \dots, A_n$ we have
$$
\mu^\ast \left( \bigcup_{k=1}^n A_k \right) = \sum_{k=1}^n \mu^\ast(...
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Lebesgue outer measure additivity
I have just showed that if $\{I_k \}_k $ is a collection of disjoint open intervals of $\mathbb{R}$ that $$ \lambda ^* \left( \bigcup_{k=1}^{\infty }I_k \right) = \sum_{k=1}^{\infty } \ell (I_k ) $$ ...
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Could we define inner measure instead of outer measure?
We all have seen outer measure in measure theory.It is the precursor of a measure.It is defined as follows:
Let $A\subset \mathbb R$,then the outer measure $\mu^*:\mathcal P(\mathbb R)\to [0,\infty]$ ...
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Completness of restricted outer measure.
$\newcommand{\scrF}{\mathscr{F}}$
Problem: Let $\mu^\ast$ be an outer measure defined on $\Omega$. Show that if $A \subseteq \Omega$ satisfies $\mu^\ast(A) = 0$ then $A$ is $\mu^\ast$-measurable. ...
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$\mathcal{L}^n(K_t) = 0 \implies \mathcal{L}^{n+1}(\cup_{t \in \mathbb{R}}{\{t\} \times K_t} ) = 0$
For any $t \in \mathbb{R}$ let $K_t$ be a measurable subset of $\mathbb{R}^n$ with $0$ Lebesgue measure. Let also
$K := \bigcup_{t \in \mathbb{R}}{\{t\} \times K_t}$
I know that if $K$ is measurable ...
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Measure and integration problem.
Suppose $A \subset{\mathbb R^{n}}$ and $\epsilon >0$.
(i) Find an open $V\supset{A}$ such that
$ m_{n}^{*}(A) \leq m_{n}^{*}(V) \leq m_{n}^{*}(A) + \epsilon $.
(ii) Does $m_{n}^{*}(V\setminus A) &...
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Show this for measurable and disjoint sets
I have problems solving this exercise:
Suppose $A,B \subset \mathbb R^{n}$ are measurable and disjoint. Show that every $E\subset \mathbb R^{n}$ satisfies
$m_{n}^{*}(E \cap (A \cup B))=m_{n}^{*}(E \...
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Is max of outer measure also an outer measure
Take $\mu^*_1,\mu^*_2$ be an outer measures on X, then prove that
$$\nu^*(A)=\max\{ \mu^*_1(A),\mu^*_2(A) \},~A\in2^X$$ is an outer measure.
Let's remind definition of outer measure.
Let $X$ be any (...
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Are the Borel sets precisely those whose measure is fixed by finite additivity?
Let $\mu:\mathscr P(\mathbb R)\to[0,\infty]$ satisfy:
(Finite additivity) $A\cap B=\varnothing\implies$$\mu(A\cup B)=\mu(A)+\mu(B)$
(Interval property) $a<b\in\mathbb R\implies\mu\big((a,b)\big)=b-...
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Outer Measure limit equality
Suppose M is the class of measurable sets with respect to an outer measure $\mu^*$ defined on the subsets of $\Omega$. Take {$E_n$} to be a monotone increasing sequence of sets in M and A any set in $\...
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Does every cover has a countable subcover?
When dealing with outer measure, I came up with this problem when extending a set function $\rho:\mathcal{C}\to [0,\infty]$ with $X=\bigcup\mathcal{C}$ and $\rho(\varnothing)=0$ to an outer measure by ...
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The additivity of outer measure for disjoint sets, one of these is Borel set.
Denote Lebesgue outer measure $\mu^\ast$.
I want to show
if $A,B\subset \mathbb R$ are disjoint and one of these sets is Borel set, then $\mu^\ast(A\cup B)=\mu^\ast(A)+\mu^\ast(B)$ holds. $\cdots \...
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Outer measure on $\mathbb N$
I was wondering if there exists a non trivial outer measure on the natural numbers $\mathbb N$.
$\mu(A) = 0$ for all $A\in\mathfrak P(\mathbb N)$
Is certainly monotonic, $\sigma$-sub additive and $\...
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Clarification regarding the proof for the Outer Measure of the set $E=(\mathbb Q\times \mathbb R) \cup (\mathbb R\times \mathbb Q)$
I was reading this proof and needed a clarification. The question is too old to post in it, hence this question.
The original link is here:
Outer Measure of the set $E=(\mathbb Q\times \mathbb R) \cup ...
user529632
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The meaning of $\mu^\ast(A)=\infty$, where $\mu^\ast$ is outer measure.
In the measure theory written by Axler, the outer measure $\mu^\ast : \mathcal P(\mathbb R)\to [0,\infty]$ is defined by $$\mathcal P(\mathbb R)\ni A\mapsto \inf \left\{ \sum_{k=1}^\infty L(I_k) \ \...
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Question about the definition of the outer measure
From Rudin's Principles of Mathematical Analysis, we define the outer measure as so:
Definition $11.7$: Let $\mu$ be additive, regular, nonnegative, and finite on $\mathcal{E}$. Consider countable ...
user529632
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Proving NonAdditivity of Outer Measure, when countable subadditivity is proven
I am self studying measure theory from Sheldon Axler's book Measure, Integration & Real Analysis. In page 21 (2.18) the author presents the non additivity of Outer Measure as:
There exist disjoint ...
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0
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Show that inner measure is the same as outer measure on Lebesgue measurable sets.
I am trying to answer part (a) problem from Axler's book, Measure, Integration and Real Analysis (seen below). I think I'm almost there but I'm stuck on a certain case.
We also have the following ...
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Let $\mu ^{*}$ be a arbitrary outer measure is necessarily satisfied? $\mu ^*(A \cup B) + \mu^{*}(A \cap B) \leq \mu^{*}(A)+\mu ^{*}(B)$
Let $\mu ^{*}$ be a arbitrary outer measure in a set $X$ and $A,B \subset X$, the following inequality is necessarily satisfied?
$$\mu ^*(A \cup B) + \mu^{*}(A \cap B) \leq \mu^{*}(A)+\mu ^{*}(B)$$
I ...
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Counterexample or proof of outer measure inequality
Let $X$ be a non empty set and $\mu^{*}:P(X) \longrightarrow [0,\infty]$ an outer measure, that is, $\mu^{*}$ satisfies the following properties:
$\mu^{*}(\emptyset)=0$;
If $A \subset B$ then $\mu^{*}...
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Showing $\pi^*$ measurability of a set, given $\pi^*$ measurability of a symmetric sum,
Suppose that $\mathfrak A$ is an algebra on $X$ and that $\mu$ is a measure on $\mathfrak A$. It is given that $\pi^*(E\triangle F)=0, F \in M$, the set of all $\pi^*$ measurable sets. Then, it is to ...
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Constructing Outer Measures Folland Proposition 1.10
On page 29 of Folland's real analysis book one can read the following proposition:
Proposition 1.10. Let $\mathcal{E} \subset \mathcal{P}(X)$ and $\rho: \mathcal{E} \mapsto [0,\infty]$ be such that $\...
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