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Math Help - finite measure space

  1. #1
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    finite measure space

    Let (X, \mathcal{B}, \mu) be a finite measure space and f be a nonnegative measurable function of X. For each A \in \mathcal{B}, set
    \nu(A)=\int_Af d\mu.

    Verify that it is a finite measure if and only if f is integrable.

    I do not see how this if and only if follows. Any hints on this would be great. Thanks in advance.
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  2. #2
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    Quote Originally Posted by eskimo343 View Post
    Let (X, \mathcal{B}, \mu) be a finite measure space and f be a nonnegative measurable function of X. For each A \in \mathcal{B}, set
    \nu(A)=\int_Af d\mu.

    Verify that it is a finite measure if and only if f is integrable.

    I do not see how this if and only if follows. Any hints on this would be great. Thanks in advance.
    - Note that \nu(X)=||f||. This is the main connection with finiteness and integrability.

    - Obviously \nu(\emptyset)=0. For the linearity, first prove it for finite union (using the fact that \mathbf{1}_{A\cup B}=\mathbf{1}_{A}+\mathbf{1}_{B}-=\mathbf{1}_{A\cap B}, then use dominated convergence to finish the result.

    Let me know if you need any more help.
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