# Rigorous treatment for continuous mass systems

I would like to ask if anyone knows an accessible, yet rigorous way of passing from a discrete system of mass-points to a continuous mass system.

For instance, we clearly know how to define the position vector of the CM (Center of Mass) of a discrete system, however, once we pass to a continuous system, we just "intuitively adapt" the definition changing the masses of mass-points into differentials (that is $$dm$$) and sums into integrals over the total mass of the system. Now, I would be fine to axiomatically re-define the position vector of the CM this way; my problem is that we did not define a function of which $$dm$$ is a differential and why should, in general, the vector position be a function of mass, and this all bothers me. Usually, when a similar passage from sums into integrals is executed, we use the properties of Cauchy sums and their links with Riemann integrals, but this time I couldn't find anything related.

As always, any opinion, hint or answer is highly appreciated!

• If you really want to be rigorous, then one way is to interpret $m$ as being a measure (in the sense of measure theory). So, for any (measurable) subset $A\subset\Bbb{R}^n$, the number $m(A)$ denotes the mass of the set $A$. Then, things like center of mass, moment of inertia and so on are all Lebesgue integrals with respect to the measure $m$. Often, such measures have a Radon-Nikodym derivative (a density) $\rho$ (as discussed in the answer below). Here is an answer I wrote detailing things for moment of inertia. Commented May 7, 2022 at 2:21
• You need this measure-theoretic approach if you wish to mathematically talk about idealized situations such as point masses, masses concentrated on lines, or planes. Or various combinations of them. Commented May 7, 2022 at 2:22

When calculating e.g. coordinates of center of mass of some continuous body, summation over particles is replaced by integration over the physical space volume. Then, symbol $$dm$$ isn't a differential of some function of position in that space; it is just a shorthand for product of density and differentials of spatial coordinates:
$$dm = \rho~ dxdydz.$$
The hypothetical function $$m(x,y,z)$$ is not important. Such function can be determined as a curiosity in one-dimensional cases, when there is only single direction and cumulative mass function along this direction $$m(x)$$ can be defined and inverted, but in 2D and 3D cases there isn't even such direction and there is no unique way to define such function.