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February 11th, 2015, 02:06 PM  #1 
Newbie Joined: Jan 2015 From: Texas Posts: 4 Thanks: 0  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 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 mathspeak. 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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2nd, 3rd, analysis, conditions, measure, measure theory, outer, prove, real analysis 
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