Radon measure as a representation of density

Question

In Applied Partial Differential Equations, chapter 8, Peter A. Markowich considers a Radon measure as the representation of the mass density of some debris.

I looked up the definition of Radon measure and its relation to other concepts but altough I have some ideas, I still don't get this.

I'm more familiar with mass represented with a density function. In this case, given a density function $\delta(x)$, we can define a measure on $\mathbb{R}^3$ by

$$\mu(A)=\int_A \delta(x) \ dx$$

so that $\mu(A)$ represents the amount of mass in the set $A$. But is this a Radon measure?

Reciprocally, given a Radon Measure in $\mathbb{R}^3$, it seems that the "density function" should be the Radon-Nikodym derivative of it with respect to the Lebesgue measure (in fact I've seen this derivative being called "density"). But of course, there are some conditions for the existence of this derivative, so I'm not sure I'm thinking correctly.

To sum up, the relation between Radon measures and densities does not seem to be straightforward. Also, I couldn't find any other text using Radon measures for densities. I'd appreciate any clarification. Thanks!

I think it's worth mentioning that there's also a relation between Radon measures and functionals. But that's a whole new story and I think it needs it's own post.

Answer 1

Suppose you want to examine the gravitational field of a circular wire. By "wire" I mean that the cross-sectional diameter is so much less than the diameter of the circle that it is negligible. It is common in such cases to treat the wire as infinitely thin, and to just use path integrals. For example, the gravitational field $\mathbf g$ at a point $\mathbf p$ is given by $$\mathbf g = \int_C \frac{G(\mathbf {r - p})}{|\mathbf{r-p}|^3}\ell\, ds$$

where $\mathbf r$ is the integration variable, $ds$ is the arclength measure, and $\ell$ is the linear density of mass on the wire. We may want to consider the case where $\ell$ is not constant.

Similarly we may want to consider the gravitation of a thin disk, $$\mathbf g = \int_D \frac{G(\mathbf {r - p})}{|\mathbf{r-p}|^3}\alpha\, dA$$ or of a ball
$$\mathbf g = \int_B \frac{G(\mathbf {r - p})}{|\mathbf{r-p}|^3}\rho\, dV$$

It would be nice to be able to unify all three cases and discuss them together instead of having to develop the same mathematics three times: once for path integration, once for surface integration, and once for volume integration (and a fourth time, for the gravitation of several discrete particles, or more times for combinations of all of them). Unfortunately, there is no volumetric density function $\rho$ that corresponds to an infinitely thin disk, wire, or particle - giving the correct result under Lebesgue or Riemman integration.

This can only be handled mathematically by expanding our concept of integration. There two common approaches to this: One approach is to generalize what is being integrated. This gives us "generalized functions" such as the Dirac delta function. The other is to express the mass distribution in the measure instead of as a density function in the integrand. This is the approach you are asking about.

In this approach we specify the mass of regions in space (i.e., of well-behaved sets), instead of the limiting ratio of mass-to-volume at points. In the three integrals above, we replace $\ell\,ds, \alpha\,dA, \rho\,dV$ with the mass-measure $dM$. The measure is defined by requiring for any set $U$ in the topological $\sigma$-algebra, $M(U)$ is the total mass in it.

We can define $M$ from a volumetric density by $$M(U) = \iiint_U \rho\, dV$$ as you've mentioned. For the disk or wire, we can define it by $$M(U) = \iint_{U\cap D}\alpha\,dA\quad\text{or}\quad M(U) = \int_{U\cap C}\ell\,ds$$ For point particles of mass $m_i$ at locations $\mathbf p_i$, we can define it as $$M(U) = \sum_{p_i \in U} m_i$$ But these are not the only possibilities for $M$. It does not need to derive from any these forms. Effectively, $M$ allows us to describe any mass distribution we want, without having to limit ourselves to particles or linear, areal, or volumnal densities. When Peter Markowich refers to the measure as being a representation of the "mass density", he is merely trying to tie the idea to a known concept. But the wording isn't explicitly correct (and I doubt he intended you to take it that way). $M$ is the mass distribution. The density is the relation of that distribution to length, area, or volume.

All of this works without restricting ourselves to Radon measures. However, just as one could invent density functions $\rho$ that are highly non-physical (for example, setting $\rho(x,y,z)$ to be the number of rational entries in $\{x, y, z\}$ - mathematically possible, but physically ludicrous), one can also come up with mass distributions $M$ that make no sense. For most things, it doesn't matter if the mass distribution is realistic or not. But allowing the worst cases limits the theorems that can be proved. Any physically viable distribution (even when allowing idealizations such as infinitely thin disks or wires) will be Radon measures, which are simply measures that respect the topology. If $\ell, \alpha, \rho$ are continuous, and the number of discrete points are finite, then each of the mass distributions defined above are Radon measures.

When the Radon-Nykodim derivative of $M$ with respect to Lebesgue measure exists, then it is indeed a mass density function $\rho$ that could be used instead. This is the purpose for which the Radon-Nykodim derivative was defined. But this concept extends the traditional idea of density exactly because it covers situations where the Radon-Nykodim derivative does not exist. The Radon-Nykodim derivative (for volume measure) does not exist for the circle, disk, and point measures defined above.

Answer 2

I think Paul Sinclair gave a great answer. I will add a bit to it by going backwards. Why do we integrate against "mass densities" in the first place? The original post writes that if $\delta$ is the density of mass of some "thing" sitting in space, then, given a reasonable subset of $A \subseteq \mathbb{R}^{3}$, it follows that the amount of mass in $A$ should be defined by $\mu(A) = \int_{A} \delta(x) \, dx$. Now $\mu$ is a set-function with a number of properties. In fact, we can prove that $\mu$ is a measure (e.g. if we restrict it to the Borel subsets of $\mathbb{R}^{3}$).

However, some reflection shows that this isn't the only measure we might want to study. For example, from the point of view of the problem we are interested in, all of the mass in $A$ might be concentrated on a line or even a point, in which case there is no way of getting a non-zero number from an expression like $\int_{A} \delta(x) \, dx$. Indeed, if the mass is concentrated at at point (say the origin), then $\mu$ should satisfy something like $\mu = M \delta_{0}$, where $\delta_{0}$ is the Dirac mass at the origin and $M$ is the total mass of the "thing" we're studying. There is no way to represent this measure by the integral of a density $\delta$ against Lebesgue measure.

The previous paragraph shows that integrals against Lebesgue measure don't capture all of the ideas we have about mass, but it also shows that many of our ideas can be accommodated with other measures. Paul Sinclair already mentions what we could do if all of our mass is concentrated on a curve or a surface. What I want to simply mention is that, in some sense, measure theory is all about mass. (Or, put another way, counting up mass, computing centers of mass, and so on is all an exercise in measure theory.) Intuitively, if I take two disjoint sets in $\mathbb{R}^{3}$, then the total mass in the two sets should be the sum of the mass in each. Thus, the mass function $\mu$, whether or not it comes from a density $\delta$, should satisfy $\mu(A_{1} \cup A_{2}) = \mu(A_{1}) + \mu(A_{2})$. The mass in the empty set should be zero so $\mu(A) = 0$. Probably the mass should be "continuous" so $\mu \left(\bigcup_{n = 1}^{\infty} A_{n}\right) = \lim_{N \to \infty} \mu(A_{1} \cup \dots \cup A_{N})$. Similarly, if $f$ tells me the amount of widgets per unit mass in space, then $\int_{\mathbb{R}^{3}} f(x) \mu(dx)$ should be the total number of widgets.

The point I'm trying to make is that if we work backwards, we see that the notion of measure corresponds quite nicely to our ideas about mass. (Mass isn't the only case where this works. We could also think about electrostatic charge, in which case we would want signed measures.) Markowich's book and Sinclair's post show that in actual physical (or "applied") problems we very well may want to work with masses other than those that are absolutely continuous with respect to Lebesgue measure (in other words, masses coming from a function $\delta$). What I am trying to convey is that in a lot of ways our conceptual (or philosophical) ideas about mass fit very well into the framework of measure theory so it's not surprising that we would use "measure" to quantify "mass" in applied problems.