Likelihood function when there is no common dominating measure?

(Not really an answer, simply an elaboration on the comment. Apologies in advance. It is a very interesting question. Hopefully the comment will be complementary to the answer(s) when they come along.)

It seems that, in order to make meaningful statistical statements, one needs densities/likelihood functions. Therefore a dominating measure necessarily shows up somewhere in the formulation, even in the non-parametric setting.

For example, take the classical fixed design non-parametric regression problem $$ y_t = f(t) + epsilon_t, ;;t = frac{k}{n}, ; k = 0, cdots, 1, $$ where $epsilon_t stackrel{i.i.d.}{sim} (0, sigma^2)$, and $f$ lies in, say, $C[0,1]$, the continuous functions on $[0,1]$.

The problem of estimating $f$ from $(y_t)$ is asymptotically equivalent to estimating drift $f$ from a sample path $Y_t$ of the stochastic process (Ito diffusion) $$ dY_t = f dt + sigma dW_t $$ where $W_t$ is standard Brownian motion. In this formulation, the problem becomes estimating a element $f$ of an infinite dimensional "parameter space" $C[0,1]$.

Statistically speaking, $Y_t$ is a probability measure $mathbb{Q}^f$ on the Skorohod space $D[0,1]$, with Radon-Nikodym density $$ frac{d mathbb{Q}^f}{ d mathbb{P}} =e^{int_0^1 frac{f}{sigma} dW_t - frac{1}{2} int_0^1 frac{f^2}{sigma^2} dt} $$ with respect to the Wiener measure $mathbb{P}$, which defines the law of $W$ (i.e. $f = 0$). This is exactly like the parametric setting, except the model ${ mathbb{Q}^f }_{ f in C[0,1] }$ has an infinite dimensional "parameter space".

I believe the notion of contiguity, introduced by Le Cam, is in similar spirit---to introduce a framework where one can speak about densities and likelihood functions, when the parameter space is not necessarily finite-dimensional.