Let $(\Omega,\Sigma,\mu)$ a measure space and $f_n:\Omega \rightarrow [-\infty;+\infty]$ measurable functions. Supposing $g\leq f_n, \qquad \forall n \in N$, a measurable function such that his negative part $g^-$ in integrable.
Show that: $ \int_\Omega \lim \inf_{n \rightarrow \infty} f_n d\mu \leq \lim \inf_{n \rightarrow \infty} \int_\Omega f_n d\mu$ and that the integral in the formula bove have sense.
This is an exercise I'm trying to solve, I started working on the demonstration that the integral have sense and then I'll try to prove the formula.
I know it's similar to the Fatou Lemma and his reverse but $f_n$ is not nonnegative and g is not dominating so I'm trying to work in that direction.
So first of all I tried to prove the existence of the limit on the right. Let $g=g^+ - g^-,\qquad g^+ - g^-\leq f_n$ so $g^-\geq g^+-f_n$ since $g^-$ is integrable I can assume that $g^+ - f_n$ is measurable too.Now can I tell that also $f_n$ is integrable as a consequence of it?
For the second integral $f_n$ in measurable,the liminf of the sucesion is measurable too but I don't know how to get to the result.