Constructing explicit elements of (L^2(μ))^∞ outside the closure of C^{∞,∞}(Y_0)

Construct an explicit element f \in L = (L^2(\mu))^\infty that does not lie in the closure of A = C^{\infty,\infty}(Y_0) with respect to the topology on L, thereby exhibiting a strict separation between L and the closure of A.

Background

The authors suspect L may be larger than A and note obstacles to approximating elements of L by A in the appropriate topology. A concrete example of an element of L not in the closure of A would confirm this disparity and clarify the relationship between analytic structures used in the amenable averaging framework.