Relation between the density function (measure theory) and density (physics)

Question

I was reading some notes on Ergodic Theory and there is this sentence:

Suppose we distribute mass on $X$ according to the mass density $fd\mu$, $f \in L^1(\mu)$,$ f \geq 0$, and then apply $T$ to every point in the space (then $T$ is some dynamical system acting on $X$). What will be the new mass distribution?

My question is, what is the relation between the density FUNCTION $f$ and the density as we study in a Physics course?

Thanks in advance :)

Answer 1

The terminology comes from physics. The quantity $f(x)$ can be interpreted as the density at the point $x$.

Answer 2

Heuristically, let $m$ be the notion of mass studied in physics, which is a measure. Let $\mu$ be the Lebesgue measure or the notion of volume in the space. The density $\rho$ is a function $\mathbb{R}^n \to \mathbb{R}$ which describes from Lebesgue measure, how to obtain the mass measure. Rigorously, it is the Radon-Nikodym derivative of mass measure to volume (Lebesgue) measure. Of course, for this density to exist, we would have to have that $m >>\mu$, but provided your mass measure is absolutely continuous w.r.t. Lebesgue measure, this is what the density is formally in Physics. There exists a function

$$\rho(x)=\frac{dm}{d\mu}$$

such that $$m(A)=\int_A\rho(x) d\mu.$$