Finding the Measure of A, B: Prove m(A)+m(B)=m(AuB)+m(AnB)

[pivoxa15]

Homework Statement

A, B in a sigma algebra

Prove
m(A)+m(B)=m(AuB)+m(AnB)

m denotes the measure.

The Attempt at a Solution

Don't see how to do it.

Somehow we are dealing with each individual set and taking the measure on them. Then finding what they equate to.

 

[morphism]

A measure is additive on countable disjoint unions. So have you tried to write A\cupB as a union of disjoint sets?

(Note: I'm assuming m is a finite measure, i.e. does not take on infinity.)

 

[Super_Leunam]

Ok you haven't tried hard Pivoxa
Hint:
use A= [A - (A \cap B)] \cup (A \cap B) \hspace{10pt}\text{or}\hspace{10pt} B= [B - (A \cap B)] \cup (A \cap B) and see what you get from it.