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Math Help - Measure theory: measure on a sigma algebra

  1. #1
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    Measure theory: measure on a sigma algebra

    Let u be a measure on a sigma algebra A of subsets of X. Prove that u(E)+u(F)=u( E\bigcup F)+u( E\bigcap F)
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  2. #2
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    Re: Measure theory: measure on a sigma algebra

    Quote Originally Posted by sawleha View Post
    Let u be a measure on a sigma algebra A of subsets of X. Prove that u(E)+u(F)=u( E\bigcup F)+u( E\bigcap F)
    $(E \cup F) = E \cup (F \cap \overline{E})$

    These are disjoint so

    $u(E \cup F) =u(E) + u(F \cap \overline{E})$

    $u(E \cup F) + u(E \cap F) = u(E) + u(F \cap \overline{E}) + u(E \cap F)$

    $F=(F \cap E) \cup (F \cap \overline{E})$

    and both of these are disjoint so

    $u(F) = u(F \cap \overline{E}) + u(E \cap F)$

    so

    $u(E \cup F) + u(E \cap F)=u(E) + u(F)$
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    Re: Measure theory: measure on a sigma algebra

    Thank you for showing each step. it really clarifies things.
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