mathematical physics - Extension to continuous in proofs of rigid body mechanics

I'm studying rigid body mechanics and I've seen several proofs of properties related to total angular momentum, kinetic energy, etc. that all regard discrete set of points. For example, to show that in an inertial frame $\frac{\text d \mathbf {L}}{\text d t}=\mathbf{\Gamma}^{\text {ext}}$, one writes down the sum

$$\mathbf{L}=\sum \mathbf{r} \times \mathbf{p},$$ and does the required derivatives.

How can such arguments be extended with rigour to continuous rigid bodies? To give an example, in "Rudin, Principle of Mathematical Analysis", in the chapter of Riemann Stieltjes integral, there's an example regarding the moment of inertia of a "straight line" body, that can be defined uniquely with the Riemann-Stieltjes integral

$$\int _0 ^{\ell} x^2\, \text dm(x).$$ Altough it's restricted to a "straight-line" body, the result is very elegant.

Back to the example of angular momentum, I think that one could define the total angular momentum as

$$\mathbf{L} =\int _{V} \mathbf {r} \times \mathbf {v(\mathbf{r})} \,\text d m(\mathbf{r}),$$ where the integral is extended over the volume of the body (actually I've never seen a definition of total angular momentum for continuous rigid bodies, since it always appears as $$\mathbf{L} = \mathbf{I}\mathbf{\omega},$$ and, of course, this is the useful identity).

So, how can the discrete arguments be extended to continuous arguments?