# Showing that a specific outer measure is a measure

• Aug 11th 2011, 03:44 PM
mgarson
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.
• Aug 11th 2011, 10:53 PM
Drexel28
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.
• Aug 16th 2011, 11:50 AM
mgarson
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 $\displaystyle \mu:P(X)\rightarrow[0,\infty]$ has to satisfy the following properties:
(i) $\displaystyle \mu(\emptyset)=0$,
(ii) $\displaystyle \mu(\cup Ej)=\sum_{1}^{\infty}\mu(E_j)$ where $\displaystyle E_i\cap E_j=\emptyset$ if $\displaystyle i\neq j$.
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Now, (i) and $\displaystyle \mu(\cup Ej)\le\sum_{1}^{\infty}\mu(E_j)$ follow from the definition of outer measure. So all we have to show is:
$\displaystyle \mu(\cup Ej)\ge\sum_{1}^{\infty}\mu(E_j)$.

And then I tried to use the following but it didn't do me any good (I'm not sure if it is true):
$\displaystyle \mu(X\setminus A_{\mu})=0 \Rightarrow X$ is countable $\displaystyle \Rightarrow X=\cup_{1}^{\infty}{x_i}$.

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 $\displaystyle \nu_{a}$ and $\displaystyle \nu_{s}$ such that: $\displaystyle \nu=\nu_{a}+\nu_{s}$ where $\displaystyle \nu_{a}<<\mu$ and $\displaystyle \nu_{s}\bot\mu$. Obviously $\displaystyle \nu(X\setminusA_{\mu})=\nu_{s}(X\setminus A_{\mu}))$ and $\displaystyle \nu_{s}(A_{\mu})=0$ since $\displaystyle \mu(X\setminus A_{\mu})$.