Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for the $L^p$-spaces defined as in Chapter III.3 of "Linear Operators, Part 1" by Dunford and Schwartz.

My question is: Is the linear span of $\{e^{ih^{*}}: h^{*}\in \mathcal{H}^{*}\}$ dense in $L^{p}(\mathcal{H},\mu)$? If not, what can be said about the closure of this span? Does it contain, for example, the bounded continuous functions on $\mathcal{H}$?

It is known that this density holds when $\mu$ is countably additive (this follows from Corollary 7.12.2 in Bogachev's "Measure Theory".)