Equivalent of Density of Stochastic Process

Consider the case where $$X$$ is a random variable defined on a space of functions say $$C[0,1]$$. Then each sample $$X(\omega)$$ is a path of stochastic process on $$[0,1]$$ as opposed to the usual case where $$X(\omega)$$ is simply a number or vector. Is it possible to have an equivalent of probability density function for such $$X$$ so that we can kind of measure probability of $$X$$ taking certain paths? For example, $$\int_Gf_X(g)dg\in \mathbb{R}$$ would be probability of $$X$$ realizes to one of the collection of paths $$G$$ where each $$g \in G$$ is a path on $$[0,1]$$.

Talking about densities requires you to fix a reference measure that your density is against. You implicitly do this when you write an integral against $$dg$$. The problem is that it's not obvious what you mean by $$dg$$.
Lemma: Let $$X$$ be an infinite dimensional normed space and let $$\mu$$ be a translation invariant measure on the Borel $$\sigma$$-algebra of $$X$$. Then $$\mu(B(0,1)) = \infty$$.
Of course, we could fix some measure on $$C[0,1]$$ and ask if there is a density against that measure but then it's not obvious what kind of measure you'd want to fix and why that would be useful here.