Help with a real analysis problem

[polkadot66]

Homework Statement

Given $f \in L^1(X, M, \mu)$ , show there is a $\sigma$ -finite measure $\mu'$ such that $\int_{E} f d\mu = \int_{E} f d\mu'$ .

Relevant Equations

$f \in L^1$ so $\int |f| d\mu < \infty$
a measure $\mu'$ is $\sigma$-finite if there are sets $A_1, A_2, ...$ such that $\mu'(A_n) < \infty$ and $X= \cup A_n$

I tried to prove this by absurd stating that there is no such $\mu'$ but i couldn't get anywhere...

 

[mathwonk]

i would begin with the definition of f being in L^1. That should give you a sequence of step functions that approximate f and can be used to define some sets of finite measure that can be used to modify mu. anyway i would do it directly, not by contradiction.