Let $\Omega \subset \mathbb R^n, n \in \mathbb N,$ be a bounded domain. Suppose $f_n \in L^1(\Omega)$ converge pointwise to a function $f: \Omega \to \mathbb R$ and $(\int_\Omega f_n)_{n \in \mathbb N}$ converges as well.
I wonder if there is a sensible way of defining $\int_\Omega f$. For instance, does the set $$ \left\{\lim_{n \to \infty} \int_\Omega f_n : L^1(\Omega) \ni f_n \to f \text{ pointwise and } \left(\int_\Omega f_n\right)_{n \in \mathbb N} \text{ converges}\right\} $$ contain in general at most one element for a given function $f$? If not, is there another way of definitng $\int_\Omega f$?
Note that I neither require $f \in L^1(\Omega)$ nor $f_n, f \ge 0$.
You have the "chasing the bump" problem. It is fairly easy to write a sequence of functions all integrating to 1 whose supports push out to the edge of the region. These will converge pointwise to zero but all integrate to 1.