1. ## 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 $\displaystyle F$?

2. Originally Posted by GTO
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?