Let $g$ integrable function on $E$ measurable set. Let $(f_n)$ measurable functions and $|f_n|\leq g$ for all $n$. Show that $\int_{E} \liminf f_n\leq \liminf \int_{E} f_n\leq \limsup \int_{E} f_n\leq \int_{E}\limsup f_n$.
I have a doubt.
with $(f_n+g)$ and Fatou, $\int \liminf (f_n+g)\leq \liminf \int (f_n+g)$
Now. Is it true that $ \liminf (f_n+g)= (\liminf f_n)+g$? I ask this, well, if it's true then $ \int \liminf (f_n+g)= \int [(\liminf f_n)+g]=\int\liminf f_n+\int g$ and $\liminf \int (f_n+g)=\liminf (\int f_n+\int g)=\liminf \int f_n+\int g$ and so $\int \liminf f_n+\int g\leq \liminf \int f_n+\int g$, and $g$ integrable, implies $\int \liminf f_n\leq \liminf \int f_n$