Showing that the set of finite $\mu$-measurable sets is a ring

Question

$\mathcal{M}_{\mathcal{e}}(\mu)$ is the set of finite $\mu$-measurable sets.

$A$ is $\mu$-measurable if a $B_n \in \mathcal{M}_{\mathcal{e}}(\mu)$ exists such that $A=\bigcup_{n=1}^\infty B_n$. We write $\mathcal{M}(\mu):=$ $\{A \subset \mathbb{R}^d | A$ is $\mu$-measurable$\}$.

How do I show that $\mathcal{M}_{\mathcal{e}}(\mu)$ is a ring?

Answer 1

You need to prove 2 things:

1. $A,B\in \mathcal{M}_e(\mu) \implies A\cup B \in \mathcal{M}_e(\mu)$
2. $A,B\in \mathcal{M}_e(\mu) \implies A\setminus B \in \mathcal{M}_e(\mu)$

In fact, $\mu(A\cup B) = \mu(A)+\mu(B)-\mu(A\cap B)<\infty$ and $\mu(A\setminus B) = \mu(A)-\mu(A\cap B)<\infty$.