My Math Forum How to prove the 2nd & 3rd conditions of outer measure?
 User Name Remember Me? Password

 Real Analysis Real Analysis Math Forum

 February 11th, 2015, 01:06 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 $\mu^*(A) = \inf \{\mu(B) \mid A \subset B, B \in \mathcal A\}$ 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.

 Tags 2nd, 3rd, analysis, conditions, measure, measure theory, outer, prove, real analysis

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post walter r Real Analysis 0 October 9th, 2014 09:40 AM mared Algebra 3 August 26th, 2014 01:23 PM goedelite Real Analysis 2 May 4th, 2012 06:42 AM guynamedluis Real Analysis 4 September 15th, 2011 05:03 PM dimper129 Linear Algebra 0 October 15th, 2009 02:09 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top