$\sigma$-finite measure $L^p$ space is isometric to a finite measure $L^p$ space

I have a measure space $(X,M,\mu)$ where $\mu$ is $\sigma$-finite. Then there exist a finite measure $\lambda$ s.t. the space $L^p(\mu)$ is isometric to $L^p(\lambda)$.

First I proved that there exist a function $h$ in $L(\mu)$ which is always positive. Then the problem is how to find the operator, because I can multiply for the inverse of $h(x)$, but how to define this measure? I tried $\lambda(A)= \int_{A} h(x) d\mu$ but then how can I send functions in the first space to the other?