The unit sphere of a nonzero subspace of a separable space is itself separable, so $growth$ against it is a countable infimum: this is the ingredient that makes $g_{A(ω)}(V)$ measurable for a merely strongly measurable $A$ (measurable in the strong operator topology, i.e. $ω ↦ A(ω)x$ measurable for each fixed $x$).