Weak star convergence question

Question

Let $C$ be a convex cone in $L^{\infty}$, that is if $x,y \in C$ and $\alpha, \beta > 0 $ then $\alpha x + \beta y \in C$. Let $U$ be the unit ball in $L^{\infty}$. Assume that for each sequence $(f_n)_{n\geq1}$ in $C \cap U$ which converges in probability to a function $f_0$ we have that $f_0 \in C \cap U$. Show $C \cap U$ is weak star closed.

This is stated in a paper and says to use Krein-Smulian and the fact that the unit ball in $L^{\infty}$ under the weak star topology is an Eberlein Compactum: a topological space homeomorphic to a subset of a Banach space in the weak topology. I can't proceed because I have no idea what to do with this convergence in probability within the functional analysis. Any help would be appreciated, thanks.

Answer 1

Convergence in $L^1$ implies convergence in probability. Therefore, $C\cap U$ is closed in $L^1$. Since, $C\cap U$ is convex, it is also weak-closed in $L^1$. Now, since the inclusion $(L^\infty,\omega^*)\rightarrow(L^1,\omega)$ is continuous, we have that $C\cap U$ is $\omega^*$ closed.

Finally, by applying Krein-Smulian Theorem we obtain the result.