I am interested in the relation between a statistical model $(\Omega, \mathcal{F}, (\mathbb{P}^\theta : \theta\in\Theta))$, where the hypotheses are "$\mathbb{P}^\theta$ is a good approximation of the real probability density" and statistical learning theory, where given input data $X:\Omega \rightarrow \mathbb{R}^m$ and labels $Y:\Omega \rightarrow \mathbb{R}^n$ we search for a function $f$ minimizing the risk $$\mathcal{E}(f)=\int_\Omega (f(X(\omega)) - Y(\omega))^2 ~\text{d}\mathbb{P}(\omega),$$ leading to hypotheses "$f$ is a good approximation of the relation between X and Y"

Are the two approaches equivalent? Can I reformulate the statistical learning approach to a statistical model, thus obtaining a set of probability measures $\mathbb{P}_f$ on some probability space, or are there fundamental differences?

I haven't found a resource where they rigorously actually do this. Any help or references are well appreciated.