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Thread: signed measure

  1. #1
    GTO
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    signed measure

    For nonnegative $\displaystyle f,g \in L^1(X,F,\mu)$ and sub$\displaystyle \sigma$algebra $\displaystyle \Lambda$ which is not a subset of $\displaystyle F$, define $\displaystyle v(A)=\int_A (f-g) d\mu$ for all $\displaystyle A \in \Lambda$.
    Show that $\displaystyle v$ is a signed measure.

    i got confused here. since $\displaystyle \Lambda$ is not a subset of $\displaystyle F$, how do i find the integral? can i integrate it only when $\displaystyle A$ is an element of $\displaystyle F$?
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  2. #2
    GTO
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    Quote Originally Posted by GTO View Post
    For nonnegative $\displaystyle f,g \in L^1(X,F,\mu)$ and sub$\displaystyle \sigma$algebra $\displaystyle \Lambda$ which is not a subset of $\displaystyle F$, define $\displaystyle v(A)=\int_A (f-g) d\mu$ for all $\displaystyle A \in \Lambda$.
    Show that $\displaystyle v$ is a signed measure.

    i got confused here. since $\displaystyle \Lambda$ is not a subset of $\displaystyle F$, how do i find the integral? can i integrate it only when $\displaystyle A$ is an element of $\displaystyle F$?
    If $\displaystyle A$ is not in $\displaystyle F$, then what is $\displaystyle v(A)=\int_A (f-g) d\mu$? is it zero or undefined?
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