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Thread: Symmetric difference

  1. #1
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    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.
    Last edited by detalosi; Sep 26th 2019 at 01:20 AM.
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  2. #2
    MHF Contributor
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    Re: Symmetric difference

    Show that

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