This is a "soft" question, in the sense that I'm looking for historical remarks and general commentary rather than a definite answer.

For compact $X \in R^n$ and $f : R^n \to R$ consider the problem $$ \min_{x \in X} ~ f(x). $$

A common trick that I've stumbled upon in several applications is to look at a "probabilistic" but equivalent reformulation $$ \min_{\mu \in P(X)} \int f(x) \, \mathrm{d}\mu(x), $$ where $P(X)$ denote the set of probability distributions over $X$.

I'm interested in history/old references on this "lifting" principle, where the idea is to artificially enlarge the solution space in order to gain some favorable properties (such as linearity or convexity).

A popular example is the linear programming formulation of optimal transport due to Kantorovich, but I wonder where this idea of "probabilistic reformulation" originated or was first used (maybe in statistical physics)?

Another example of this trick are policy gradient methods in reinforcement learning, for an explanation see this blog post: http://www.argmin.net/2018/02/20/reinforce/