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Math Help - Finite additive measure is a measure if it is continuous from below

  1. #1
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    Finite additive measure is a measure if it is continuous from below

    Suppose that a finite additive measure  \mu : \mathbb {M} \rightarrow [0, \infty ] is continuous from below, I need to show that it is a measure.

    Proof.

    Let  \{ E_n \} ^ \infty _{n=1} \subset \mathbb {M}, all disjoint.

    Define F_n= \bigcup ^n _{j=1}E_j.

    Note that: 1.  \bigcup _{j=1} ^ \infty F_j = \bigcup _{j=1}^ \infty E_j and 2.  F_j \subset F_{j+1}

    Then we have  \mu ( \bigcup _{j=1}^ \infty F_j) = \lim _{n \rightarrow \infty } \mu (F_n)

    Now,  \mu ( \bigcup _{j=1}^ \infty E_j ) = \mu ( \bigcup _{j=1}^ \infty F_j ) = \lim _{n \rightarrow \infty } \mu (F_n) = \lim _{ n \rightarrow \infty } \mu ( \bigcup _{j=1}^n E_j ) = \lim _{n \rightarrow \infty } \sum _{j=1}^n \mu (E_j) = \sum _{j=1}^ \infty \mu (E_j) .

    Is this correct? Thank you.
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  2. #2
    Moo
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    Looks correct
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