L ∞ Spaces of Vector-Valued Functions as Spaces of Continuous Functions

It is proved that for any decomposable perfect measure space (𝑍, 𝒜, 𝜇), the space 𝐿𝜔∗∞ (𝜇, 𝐸*) of essentially bounded weak* measurable functions on 𝑍 to 𝐸* is linearly isometric to the space 𝐶(𝑍,𝐸∗*) of continuous functions on 𝑍 to 𝐸∗*, the latter space is being provided with the supremum norm ‖𝑔‖∞ = sup𝑧∈𝑍‖𝑔(𝑧)‖⁡ where 𝐸∗* stands for the space 𝐸* endowed with its weak* topology.