Before revision:  Second, we provide some relations between the Fatou property and some other types of lower semicontinuity, namely norm lower semicontinuity and the strong Fatou property. In particular, we show that when \mathcal{X\neq L^1, the strong Fatou property and the Fatou property coincide.
After revision:  Second, we provide some relations between the Fatou property \neq L^1, the strong Fatou property and the Fatou property coincide.