# 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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### 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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### 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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### $m^\star(A\cup B)<m^\star(A)+m^\star(B)$

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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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 $\... • 594 0 votes 0 answers 33 views ### Clarification regarding the proof for the Outer Measure of the set$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 ... 52 views

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