pr.probability - A geometric interpretation of independence?

If you leave the realm of abstract probability spaces and focus on probability in Banach spaces, there's a lot of geometry to take advantage of. Here's an example.

Let $X$ be a Banach space, and let $mathbb P$ be a Radon probability measure on $X$ such that continuous linear functionals are square-integrable (i.e. $int_X |f(x)|^2 ~dmathbb P(x) < infty$ for all $f in X^*$). For example, $X = C([0,1])$ with Wiener measure $mathbb P$.

These are sufficient conditions for there to exist a mean $m in X$ and covariance operator $K : X^* to X$ such that $$mathbb Ef = f(m) qquad mathrm{and} qquad mathbb E (fg) - f(m)g(m) = f(Kg)$$
for all $f, g in X^*$. One can show that

$$mathbb P left( m + overline{KX^*} right) = 1.$$

Under these very general assumptions, the probability concentrates on the affine subspace generated by the mean and covariance.

_{Reference: Vakhania, Tarieladze and Chobanyan, Probability Distributions in Banach Spaces}