# 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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### Does my rigorous definiiton make sense and give what I want? How do we simplify my definiton?

My previous measure in this post doesn't make sense so I made modifications. I need someone to check whether my definitions improved. Suppose we have the following definition? Definition $\ell$ is ...
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### Rings of Continuous functions [closed]

In a ring $C(\mathbb R)$, the ideal $O_0$ of all functions that vanish on a neighbourhood of $0$ is a prime ideal?
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### Why the Lebesgue outer measure of the boundary of rectangle in $\mathbb{R^n}$ is zero?

Let $A$ be a closed rectangle in $\mathbb{R^n}$ and let $m^*$ be Lebesgue outer measure. And let $\partial A$ be the boundary of $A$. Then, prove that $m^* (\partial A)=0.$ Since $A$ is a closed ...
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### Prob. 26, Chap. 2, in Royden's REAL ANALYSIS: Proof of $m^*\left(A\cap\bigcup_{k=1}^\infty E_k\right)=\sum_{k=1}^\infty m^*\left(A\cap B_k\right)$

Here is Prob. 26, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $\left\{ E_k \right\}_{k=1}^\infty$ be a countable disjoint collection of measurable sets. ...
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### Example of strictly additive outer measure

I am having a difficulty in understanding the proof of "strict sub-additivity" of outer measure. While I took the help of internet, here I found the same example that my teacher gave on his ...
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### Positive measure of a Lebesgue measurable set

Here E is a Lebesgue measurable set in $R$. Show if the following is true or false: Every uncountable measurable sets must have positive measure. Every set with positive outer measure is Lebesgue ...
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### Prob. 19, Chap. 2, in Royden's REAL ANALYSIS: For a nonmeasurable set $E$ of finite outer measure there exists an open set $O \supset E$ such that …

Here is Prob. 19, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $E$ have finite outer measure. Show that if $E$ is not measurable, then there is an open ...
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### Give an example of failure of the continuity of outer Lebesgue measure

We know that Lebesgue Measure possesses the following continuity property: If ${\{B_k\}_{k=1}^{\infty}}$ is a descending collection of measurable sets and $m(B_1)<\infty$ , then \begin{equation} m\...
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### Prob. 8, Chap. 2, in Royden's REAL ANALYSIS: The collection of finitely many open intervals covering the rational numbers in $[0, 1]$ …

Here is Prob. 8, Chap. 2, in the book Real Analysis by H. L. Royden and P. M. Fitzpatrick, 4th edition: Let $B$ be the set of rational numbers in the interval $[0, 1]$, and let \$\left\{ I_k \right\}_{...