Weak topology and properties fulfilled almost everywhere

Let $B$ be a Banach space. A sequence of $B$-valued functions $\langle f_n\rangle$ is weakly almost everywhere convergent to $0$ provided $x^*\circ f_n$ is almost everywhere convergent to $0$ for every continuous linear $x^*$ on $B$. A Banach space is finite dimensional if and only if every weakly almost everywhere convergent sequence of $B$-valued functions is almost everywhere bounded. If $B$ is separable, $B^*$ is separable if and only if every weakly almost everywhere convergent to $0$ and almost everywhere bounded sequence of $B$-valued functions is weakly convergent to $0$ almost everywhere.