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        riddles >> medium >> outer measure
        
(Message started by: MonicaMath on Sep 27^th, 2009, 9:38pm)

Title: outer measure
Post by MonicaMath on Sep 27^th, 2009, 9:38pm

is there an example for a disjoint sequense of sets {A_k} that satisfies the countably subadditive (strictly < ) property of outer measure ??

Title: Re: outer measure
Post by Eigenray on Sep 28^th, 2009, 12:26am

A simple example is to take any set with more than one element, and define the outer measure of any non-empty set to be 1.

If you mean an example for the Lebesgue outer measure, consider a [link=http://en.wikipedia.org/wiki/Vitali_set]Vitali set[/link] V. All translations of V have the same outer measure c, so countable subadditivity implies c > 0. But the sum is then infinite, giving strict inequality, no matter the actual value of c (which depends on the choice of V).

Title: Re: outer measure
Post by MonicaMath on Sep 28^th, 2009, 11:31pm

OK, thank you.
But I'm wondering .... does the Vitali set has measure 1 ?? and if so does this contradicts that it is a subset of [0,1] which has measure 1 also?

thank you

Title: Re: outer measure
Post by Eigenray on Sep 29^th, 2009, 4:01am

on 09/28/09 at 23:31:38, MonicaMath wrote:

OK, thank you.
But I'm wondering .... does the Vitali set has measure 1

The Vitali set V is not measurable. The union of countably many disjoint translates of V has positive, finite measure, so there is no way for countable additivity to hold. But V is outer measurable. It has positive outer measure, which demonstrates strict subadditivity.

The actual outer measure of V depends on V, but it can be any number in (0,1]. See [link=http://groups.google.com/group/sci.math/msg/bbf375f519887c23]this post[/link].

Quote:

and if so does this contradicts that it is a subset of [0,1] which has measure 1 also?

I don't know what you mean. There are many subsets of [0,1] which have measure 1, for example [0,1] with countably many points removed. More generally one can remove any set of measure 0, like the Cantor set.