This is a question that I came across while learning measure theory. As a matter of fact I am completely new to this area so I do not really know how to approach it. Some hints or perhaps guidelines to solve this questions will be very appreciated.
Let $f\in L^+(X,M)$ and let $\mu$ be a positive measure on $M$. For $E\in M$, define $\lambda(E)=\int _{E}f d\mu$. Using the fact that $\lambda$ is a measure on $E$, show that for all $g\in L^+(X,M)$, $\int g(x)d\lambda (x)$=$\int f(x)g(x)d\mu (x)$