  In this paper, we explore several Fatou-type properties of risk measures. The
paper continues to reveal that the strong Fatou property, which was introduced
in [17], seems to be most suitable to ensure nice dual representations of risk
measures. Our main result asserts that every quasiconvex law-invariant
functional on a rearrangement invariant space $\mathcal{X}$ with the strong
Fatou property is $\sigma(\mathcal{X},L^\infty)$ lower semicontinuous and that
the converse is true on a wide range of rearrangement invariant spaces. We also
study inf-convolutions of law-invariant or surplus-invariant risk measures that
preserve the (strong) Fatou property.
