Another example from theoretical high-energy physics I've encountered: sometimes when physicists have some equation of motion for an arbitrary number $N$ of particles with positions $x_i$, e.g. something of the form $\frac{1}{N}\sum_i f(x_i) + \frac{1}{N^2}\sum_{ij} g(x_i, x_j) = 0$, they wish to know what the solutions to this equation look like for large $N$. A technique they use is to replace the variables $x_i$ with a probability measure $\mu$ on the space of their possible values, which is supposed to represent the number of $x_i$'s in a given region in the large $N$ limit, and instead of solving the original equation they solve the analogous equation in $\mu$, e.g. $\int f(x) \mathrm{d}\mu(x) + \int g(x, y) \mathrm{d}(\mu \times \mu) (x, y) = 0$. In fact it's not hard to come up with a toy example where the original equation can be solved exactly for all $N$ and the solutions "look like" a particular probability distribution in the large $N$ limit, but that probability distribution fails to satisfy the corresponding equation, and for that reason I have some doubt that this method can be turned into something rigorous.