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Math Help - Sup and inf of bounded variations

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    Sup and inf of bounded variations

    Suppose that v is a signed measure on  (X, \mathbb {M} ) and  E \subset \mathbb {M}
    Prove that:
    a)  v^+(E) = sup \{ v(F) : F \in \mathbb {M} , F \subset E \}

    b)  v^-(E) = -inf \{ v(F) : F \in \mathbb {M} , F \subset E \}

    c)  \mid v \mid (E) = sup \{ \sum _1 ^n \mid v(E_j) \mid : n \in \mathbb {N} , E_1,...,E_n \ disjoint, \ \bigcup _1 ^n E_j = E \}

    Proof so far.

    a) Now v(E) = v^+(E)-v^-(E) , so we have  v(E) \leq v^+(E) \ \ \ \forall E
    Since v^+ \bot v^- , we have  X = P \cup N , P and N are disjoint, and N is v^+ -null, P is v^--null.

    If I pick F \subset P , then I will have v(F)=v^+(F)-v^-(F)=v^+(F).

    And that proves (a). (b) should be similar I think.

    But I'm a bit lost on (c), mainly because of the sums there.

    If I rewrite  \mid v \mid (E) = v^+(E)+v^-(E)= sup \{ v(F) : F \in \mathbb {M} , F \subset E \} -inf \{ v(F) : F \in \mathbb {M} , F \subset E \} , will it help?

    Thank you.
    Last edited by tttcomrader; December 12th 2009 at 06:26 PM.
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