Before revision:  First , we show that for any proper quasiconvex law-invariant functional \rho on \mathcal{X \sigma(X,X_n^\sim) -lower semicontinuity and \sigma(X,L^\infty) -lower semicontinuity of \rho are equivalent, where \mathcal{X}_n^\sim is the order continuous dual of \mathcal{X}.
After revision:  In particular , we show that for any proper quasiconvex law-invariant functional \rho on \mathcal{X \sigma(X,X_n^\sim) -lower semicontinuity and \sigma(X,L^\infty) -lower semicontinuity of \rho are equivalent, where \mathcal{X}_n^\sim is the order continuous dual of \mathcal{X}.