I'm currently working out exercise 3.6 from Bass' Real analysis for graduate students. The problem is as follows:
Prove that if (X, A , µ) is a measure space, B ∈ A, and we define ν(A) = µ(A ∩ B) for A ∈ A, then ν is a measure.
I understand the concept behind a σ-algebra, and the relation between the measureable set and µ. However, I have hit a roadblock when trying to apply the information to this problem.
My issue is mainly figuring out where to start on this problem. I initially tried changing A ∩ B to (A' ∪ B')' ( where ' denotes the complement of a set) in order to expand this out/attempt to simplify it in some way. However, I am not sure I'm quite on the right track.
Any hints/tips you could give me would be much appreciated. Also, here is Bass' book for reference.
We have to prove that 1) $v(\emptyset)=0$; 2)$v$ is positive and 3)$v$ is countably addictive. I think you know how to do 1 and 2... For 3:
Given a sequence $(A_i)$ of disjoint sets , $v(\bigcup\limits_{i=1}^{\infty}A_i)=\mu(B\cap\bigcup\limits_{i=1}^{\infty}A_i)=\mu(\bigcup\limits_{i=1}^{\infty}B\cap A_i)$ and how $ B\cap A_i$ is a family of disjoints, we get the desired.