Follow up to this question: Is there such a thing as a "dual measure"?
Let $V$ be a finite dimensional vector space and $X : \Omega \to V$ a random vector in $L^2$ (this is maybe fuzzy)
The pushforward $\mu := X_* \mathbb{P}$ defines a measure on $V$.
This measure induces an inner product on $V^*$:
$$ \langle f, g \rangle = \int f(v)g(v) \mathrm{d} \mu(v)$$
There's a natural dual inner product on $V$ that makes the Riesz representation into an isometry. (The distance coming from this inner product is used to define the Gaussian for instance)
Is there a kind of "dual measure" on $V^*$ that induces the dual inner product on $V$ as seen above?