Abstract Wiener space

Remarks

The measure $\mu$ is unique, and it is called the abstract Wiener measure; it is a centered Gaussian measure. Equivalently, the image under $i$ of the canonical Gaussian cylinder set measure of $H$ extends to a countably additive measure on $E$. Conversely, every centered Gaussian measure on a separable Banach space arises in this way, with $H$ its Cameron–Martin space.

Example: let $E = C_0([0, 1])$ be the classical Wiener space with the sup norm. Let $H$ be the space of absolutely continuous $h \in E$ with $h' \in L^2$, with $\ip{h}{k} = \int_0^1 h' k' \, dt$, and let $i$ be the inclusion. With $\mu$ the Wiener measure, take $\ell(\omega) = \omega(t)$. Then $i^* \ell$ is $s \mapsto \min(s, t)$, whose norm squared is $t$, and indeed $\omega(t)$ has law $N(0, t)$.

Non-example: let $E = H$ be infinite-dimensional and $i$ the identity. For an orthonormal sequence $(e_k)$, the readings $\ip{x}{e_k}$ would be independent standard normal variables under $\mu$. Then $\sum_k \ip{x}{e_k}^2 = \infty$ almost surely, which is impossible for $x \in H$, so no such $\mu$ exists.