Problem related to signed measure

[onthetopo]

Homework Statement

(X,S,u) a measure space and f is in L1.
Show that for any e>0, there exists a set E with u(E)<+infinity such that
$$| \int_{E} fdu - \int_{X} fdu |<e$$

The Attempt at a Solution

we can define a function
$$v(A)=\int_{A}fdu$$
It is a well known result that v(A) is in fact a signed measure.

We can somehow use the property of signed measures to show that there always exist a E such that |v(E)-v(X)|<e?

 

[morphism]

Don't you need sigma-finiteness here or something of the sort?

 

[onthetopo]

The question is as is, there is no mention of sigma finiteness. My attempt at the solution could be completely wrong.

 

[grossgermany]

Let's suppose it is sigma finite? then what?