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September 14th, 2011, 12:20 AM   #1
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Outer Measures

Problem:
Let denote the length of an open interval .
A covering of a subset by open intervals is a collection of nonempty open intervals , indexed by some set , such that .
The outer measure of a subset is is a covering of by open intervals, where we adopt the convention that if is uncountable

What is ?


Here are my thoughts:
Since 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 , then does that imply that the outer measure of is . Does this follow?
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September 14th, 2011, 02:55 PM   #2
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Re: Outer Measures

It would clarify things if you would define Q for someone not familiar with the notation.
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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.
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September 15th, 2011, 01:03 PM   #4
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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}.
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September 15th, 2011, 05:03 PM   #5
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Re: Outer Measures

I actually solved this the same way you describe this morning.
Thanks for the help
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