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February 11th, 2015, 02:06 PM   #1
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How to prove the 2nd & 3rd conditions of outer measure?

I have this question on outer measure from Richard Bass' book, supposed to be an introductory question, but I am lost:
Prove that $\mu^*$ is an outer measure, given a measure space $(X, \mathcal A, \mu)$ and define

$\mu^*(A) = \inf \{\mu(B) \mid A \subset B, B \in \mathcal A\}$

for all subsets $A$ of $X$.

Here are what I have gone so far:
(1) The first condition is the easiest one:

$\begin{align}
\mu^*(\emptyset) &= \inf \{\mu(B) \mid \emptyset \subset B, B \in \mathcal A\}\\
&= \mu (\emptyset) \\
&= 0
\end{align}$

(2) Now the second condition. Let $D, E \in X$ and $D \subset E$,

$\begin{align}
\mu^*(D) &= \inf \{\mu(D') \mid D \subset D', D' \in \mathcal A\}\\
\mu^*(E) &= \inf \{\mu(E') \mid E \subset E', E' \in \mathcal A\}\\
\end{align}$

Here I need to prove $\mu^* (D) \leq \mu^*(E)$. It looks to me so intuitive especially if I draw Venn diagrams of $D, E, D'$ and $E'$, but I don't know how to say it in math-speak. I would appreciate helps on this 2nd. condition.

(3) And this 3rd. condition is my major stumbling block: Given $(A_i)_{i \in \mathbb N} \subset X$, I need to arrive at

$\mu^* (\bigcup _{i=1}^{\infty} A_i)\leq \sum_{i=1}^{\infty} \mu^* (A_i).$

Here, I know for sure I need to state this first: $\forall A_i, \exists B_i $ such that $ A_i \subset B_i, B_i \in \mathcal A$, but I don't think the next step is right:

$\begin{align}
\mu^*(\bigcup_{i=1}^{\infty}A_i) &= \inf \{\bigcup_{i=1}^{\infty}\mu(B_i) \mid A_i \subset B_i, B_i \in \mathcal A\}\\
&= \ldots\\
\end{align}$

I would appreciate any help on this 3rd. condition in addition to the 2nd. above. Thank you for your time and effort.
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