# Measure Theory Question

• Sep 25th 2008, 04:43 PM
JohnStaphin
Measure Theory Question
I was told that you can find a disjoint sequence of sets...say $\displaystyle {Ei}$ such that

$\displaystyle m*(U Ei)$ < Σ $\displaystyle m*(Ei)$.. That is the measure of the union of all these sets is less than the sum of the individual measure of each set.... This is obvious if the sets aren't disjoint...But can someone give me an example of this? Thanks.
• Sep 25th 2008, 10:37 PM
Moo
Hello,
Quote:

Originally Posted by JohnStaphin
I was told that you can find a disjoint sequence of sets...say $\displaystyle {Ei}$ such that

$\displaystyle m*(U Ei)$ < Σ $\displaystyle m*(Ei)$.. That is the measure of the union of all these sets is less than the sum of the individual measure of each set.... This is obvious if the sets aren't disjoint...But can someone give me an example of this? Thanks.

Hmmm this is weird, because one of the basic properties of the measure is :

If m is a measure then :
$\displaystyle \forall \text{ sequence } (A_n)_{n \in \mathbb{N}} \text{ in the } \sigma-\text{algebra } \mathcal{A}, ~\text{where } \forall m,n ~(m \neq n), ~ A_m \cap A_n=\emptyset$ $\displaystyle , \text{ we have } m \left(\bigcup_n A_n \right) {\color{red}=} \sum_n m (A_n)$

Am I misunderstanding your question ?
• Sep 26th 2008, 03:36 AM
JohnStaphin
Yeah...Apparently E is not a measure, but then again... it doesn't make much sense to talk about m*(E) in this case... but who knows...perhaps you could show this for some non measurable set...Although I don't think you can analytically...