My Math Forum Outer Measures

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 September 14th, 2011, 12:20 AM #1 Member   Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0 Outer Measures Problem: Let $|I|$ denote the length $b-a$ of an open interval $I= (a,b)$. A covering of a subset $E \subseteq \mathbb{R}$ by open intervals is a collection $\{ I_{\alpha} \}_{\alpha \in A}$ of nonempty open intervals $I_{\alpha}$, indexed by some set $A$, such that $E \subseteq \bigcup_{\alpha\in A} I_{\alpha}$. The outer measure of a subset $E \subseteq \mathbb{R}$ is $m^*(E)= \inf\left\{ \sum_{\alpha \in A} |I_{\alpha}| \,:\, \text{I_{\alpha}$ is a covering of $E$ by open intervals$\}$, where we adopt the convention that $\sum_{\alpha \in A} |I_{\alpha}|= \infty$if $A$ is uncountable What is $m^*(\mathbb{Q})$? Here are my thoughts: Since $\mathbb{Q}$ is countable, then the covering must be countable or uncountable. Does this mean the outer measure is infinity? Another thought is that if we just focus on all positive rationals and find a finite outer measure for them, call it $L$, then does that imply that the outer measure of $\mathbb{Q}$ is $2L$. Does this follow?
 September 14th, 2011, 02:55 PM #2 Global Moderator   Joined: May 2007 Posts: 6,805 Thanks: 716 Re: Outer Measures It would clarify things if you would define Q for someone not familiar with the notation.
September 14th, 2011, 04:05 PM   #3
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Re: Outer Measures

Quote:
 Originally Posted by mathman It would clarify things if you would define Q for someone not familiar with the notation.
Sorry. It is the symbol for the set of all rational numbers.

 September 15th, 2011, 01:03 PM #4 Global Moderator   Joined: May 2007 Posts: 6,805 Thanks: 716 Re: Outer Measures Since Q is countable, we can always construct a countable covering, so getting an inf is always possible. In particular consider some ordering of Q. Cover the nth member by an an interval of length x/2^n. The sum of all the intervals will be x. Since x can be arbitrarily small, the outer measure is 0, i.e. inf {x|x>0}.
 September 15th, 2011, 05:03 PM #5 Member   Joined: Dec 2010 From: Miami, FL Posts: 96 Thanks: 0 Re: Outer Measures I actually solved this the same way you describe this morning. Thanks for the help

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