Weak compactness of nonnegative part of unit ball of $L^1$

Question

Let $(\Omega,\mathfrak{A},\mu)$ be a measure space and let $B$ be the unit ball of $L^1(\Omega,\mathfrak{A},\mu;\mathbb{R})$. Suppose that $$ B_+=\big\{u\in B\;|\;u\geq 0\;\:\text{$\mu$-a.e.}\big\} $$ is weakly compact.

My question: Does the weak compactness of $B_+$ imply the weak compactness of $B$? I believe that it is the case, but I have no clue where to start.

Any help/hint is highly appreciated.

Answer 1

If $(u_i)$ is a net in $B$ then $(u_i^{+},u_i^{-})$ is a net in $B_{+} \times B_{+}$ which is weakly compact. Hence there is a convergent subset and this gives a convergent subnet of $(u_i)$.