Associate space with respect to a semi-finite measure

If μ is a semi-finite measure and E an arbitrary saturated Banach function space with respect to μ, we prove that E× is equal to the space En∗ consisting of order continuous linear functionals defined on E if, and only if, E× has the Fatou property. Additionally, if the measure μ is semi-finite and not localizable, we show that the above equality may no longer be true.