1. ## Symmetric difference

Hello,

I am considering the finite measure space $\displaystyle (X,\mathcal{A},\mu)$.

I have shown that $\displaystyle \vert \mu(A)-\mu(B) \vert \leq \mu(A \triangle B)$

I am now considering two sequences $\displaystyle (A_{n})_{n\in\mathbb{N}}$ og $\displaystyle (B_{n})_{n\in\mathbb{N}}$ satisfying $\displaystyle \mu(A_{n} \triangle A_{n} ) \leq \frac{1}{3^{n}}$.

I want to show that:

$\displaystyle \left.\mid \mu \left( \cup_{n=1}^{\infty}A_{n} \right) - \mu\left( \cup_{n=1}^{\infty}B_{n} \right) \right.\mid \leq \frac{1}{2}$

Could someone give a hint?

Thanks.

2. ## Re: Symmetric difference

Show that

$$\underset{n=1}{\overset{\infty }{\cup }}A_n \triangle \underset{n=1}{\overset{\infty }{\cup }}B_n\subseteq \underset{n=1}{\overset{\infty }{\cup }}\left(A_n \triangle B_n\right)$$