Reconstructing a measure from its (absolutely continuous) marginals

Question

Let's denote by $C$ the space of continuous functions $[0,T] \rightarrow \mathbb{R}^n$ for some fixed $T>0$ and assume we have a probability measure $Q$ on the space $C$. Consider the evaluation functions $e_t : C \rightarrow \mathbb{R}^n$ given by $e_t(\alpha) = \alpha(t)$. We can then define a family of marginals (not sure if it's the right word in this context) of $Q$ as probability measures on $\mathbb{R}^n$ given as push-forwards of $Q$ by $e_t$, i.e. $$ \rho_t := e_t {}_{\#} Q \ldotp$$ Let's assume that we've shown that the $\rho_t$'s are absolutely continuous with respect to the standard Lebesgue measure. Can we somehow reconstruct $Q$ knowing only the distributions $\rho_t$? in particular, can we rewrite integrals of the form $$\int_C \int_0^T f(t,\gamma(t))dt dQ$$ as something like $$\int_0^T \int_{\mathbb{R}^n} f(t,x) \rho_t(x) dx dt \quad ?$$ Assuming $f$ is at least nice enough to apply Fubini, say positive. Basically I'm interested in what can we say about the family $\{ \rho_t\}$ and vice-versa what can we say about $Q$ knowing that its marginals have densities wrt the Lebesgue measure. Can we for example say that the function $$\rho(t,x) := \rho_t(x)$$ is, at least, measurable? Here $\rho_t(x)$ denotes the density of the measure $\rho_t$ at point $x$.