signed measure

**GTO** · January 2nd 2010, 01:34 PM

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

i got confused here. since $\Lambda$ is not a subset of $F$ , how do i find the integral? can i integrate it only when $A$ is an element of $F$ ?

**GTO** · January 4th 2010, 04:04 PM

Originally Posted by GTO

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

i got confused here. since $\Lambda$ is not a subset of $F$ , how do i find the integral? can i integrate it only when $A$ is an element of $F$ ?

If $A$ is not in $F$ , then what is $v(A)=\int_A (f-g) d\mu$ ? is it zero or undefined?

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