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.
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 .