Showing that a specific outer measure is a measure

Problems:

a. Let X be an in infinite set and let mu be a fnite outer measure on the

subsets of X such that every set {x}; x in X is mu-measurable and there

exists a countable subset A of X with mu(X\A) = 0. Show that mu is a

measure on the sigma-algebra of all subsets of X.

b) Let mu, vu be finite outer measures on X with the property in a), that is,

there exist countable sets A, B in X with mu(X\A) = vu(X\B) = 0.

Find the Lebesgue decomposition of vu with respect to mu.

Re: Showing that a specific outer measure is a measure

Quote:

Originally Posted by

**mgarson** Problems:

a. Let X be an in infinite set and let mu be a fnite outer measure on the

subsets of X such that every set {x}; x in X is mu-measurable and there

exists a countable subset A of X with mu(X\A) = 0. Show that mu is a

measure on the sigma-algebra of all subsets of X.

b) Let mu, vu be finite outer measures on X with the property in a), that is,

there exist countable sets A, B in X with mu(X\A) = vu(X\B) = 0.

Find the Lebesgue decomposition of vu with respect to mu.

We need to see some work friend. The first problem really is not that hard (and I don't mean that in a condescending way), have you given it a try? Can you show us something? Maybe then we'll help with the second part.

Re: Showing that a specific outer measure is a measure

Sure! Here it goes (I didn't want to post any of my stuff before because I didn't want to confuse anybody with my attempt). Well, here it is,

a. I realize that our has to satisfy the following properties:

(i) ,

(ii) where if .

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Now, (i) and follow from the definition of outer measure. So all we have to show is:

.

And then I tried to use the following but it didn't do me any good (I'm not sure if it is true):

is countable .

And that's pretty much it...

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b. As for this one I really have no clue. I mean I know that I'm trying to find and such that: where and . Obviously and since .