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Math Help - Integral with another measure

  1. #1
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    Integral with another measure

    Suppose that f is a non-negative integrable function. Let  \lambda (E) = \int _E f dm for all  E \in \mathbb M , the set of all  \sigma -algebra.

    Prove that  \int g d \lambda = \int fg dm \ \ \ \forall g , g non-negative integrable.

    Proof so far.

    Now, from an earlier thread, I understand that  \lambda is a measure.

    Case 1) Suppose that g is a simple function, then g= \sum ^n_{i=1} a_i 1_{E_i} , Ei disjoint.

    Then  \int g d \lambda = \int \sum a_i1_{E_i}d \lambda = \sum ^n_{i=1} a_i \lambda (E_i)

    Which gives  = \sum _{i=1}^n a_i \int _{E_i}f dm

    Now, is this integrand equals to  \int gf dm ? I think it is but I just can't seem to work it out right.

    Thanks.
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  2. #2
    MHF Contributor

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    You should simply write \int_{E_i} f dm= \int 1_{E_i} f dm, and use linearity of integral like you did before.
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