Let $(E,\mathcal E,\mu)$ be a probability space and $(f_n)_{n\in\mathbb N}\subseteq L^1(\mu)$ with $$\int f_n\:{\rm d}\mu\xrightarrow{n\to\infty}\gamma\tag1$$ for some $\gamma\in\mathbb R$.
What does it mean if we say that $f\in L^1(\mu)$ is a limit point of $(f_n)_{n\in\mathbb N}$ with respect to the $\sigma(L^1(\mu),L^\infty(\mu))$-topology (i.e. the weak topology on $L^1(\mu)$)? I clearly know the abstract definition for a general topological space, but is there an equivalent characterization available here which is easier to handle?
And why can we conclude that $\int f\:{\rm d}\mu=\gamma$?
There exist a subnet $f_{n_i}$ conveging to $f$ weakly. Hence $\int f_{n_i} g d\mu \to \int fg d\mu$ for all $g \in L^{\infty}$. Just take $g=1$ to con lude that $\int fd\mu=\gamma$.