# Approximating a discrete measure with a continuous one

In physics it is common to approximate distributions of point masses or charges with continuous distributions. To do this, one typically defines a density function by moving throughout the space a test volume containing many points. In more refined approaches, one can consider the asymptotic behaviour of the average density while enlarging the test volume.

From the point of view of measure theory — if I understand it correctly — this would mean constructing a continuous measure in $\mathbb{R}^n$ from a discrete one, so that the continuous measure can be considered, in some sense, an approximation of the discrete one.

Is there any standard, formal, way to do such a construction in measure theory?

Given the comments below, I reckon that my question above is not clear enough. Here are a few additional remarks:

1. For discrete measure, I mean a measure $\mu$ for which there exists a countable set $A\subset\mathbb{R}^n$ such that its complement is a set of measure zero, $\mu(\complement A) = 0$. Thus, if $A = \{x_1,x_2,\ldots\}$, the measure can be represented as $\mu = \sum_{k=1}^\infty a_k\delta_{x_k}$, where $\delta_{x_k}$ is the Dirac's measure centred in $x_k$. I would consider both finite and countably infinite sets, because we usually like to speak also of unbounded bodies (e.g. a line of charge).
2. As suggested by @zhoraster in the comments, the first step of the construction is naturally that of weighing (averaging, smoothing) the discrete distribution with a smooth weighing function $\varphi(x)$, normalized so that $$\int\varphi(x)\,\mathrm{d}x = 1.$$ This approach is, for instance, well described in G. Russakoff, "A derivation of the macroscopic Maxwell equations", Am. J. Phys., 38, 1188 (I recalled the existence of this paper just today).

What I think is missing, from a mathematical point of view, in the given reference and in other treatments is the following. Physicists tend to assume, more or less tacitly (see e.g. also the treatment of the macroscopic Maxwell equations given by Jackson in his Classical electrodynamics), that the continuous distribution (measure) obtainable by the smoothing process is somehow independent of the chosen weighing function, if its effective volume is large enough to contain many points. I think that the critical point is this one, not that of weighing: so, is there any standard proof in measure theory showing that through a process of smoothing we can obtain a well-defined continuous measure?

Maybe I'm overthinking all the above process and things are really much more trivial, but any clarification or pointer to references will be appreciated.

• What do you mean by a "discrete measure"? A linear combination of delta-measures $\mu = \sum_{n\ge 0} a_n \delta_{x_n}$? What is wrong with taking $\mu_N(dx) = \sum_{n: |x_n|<N} a_n N\varphi(N(x-x_n)) dx$ with $\varphi$ non-negative and $\int \varphi(x) dx = 1$? – zhoraster Nov 24 '15 at 8:33
• Gaussian distributions provide a standard way. For every $m$ in $\mathbb R^n$ and $\sigma^2$ positive, let $\gamma_{m,\sigma^2}$ denote the normal measure with density $$(2\pi\sigma^2)^{-n/2}e^{-\|x-m\|^2/(2\sigma^2)},$$ then every discrete probability measure $$\sum_kp_k\delta_{x_k}$$ can be approximated by $$\sum_kp_k\gamma_{x_k,\sigma^2},$$ when $\sigma^2\to0$. An almost sure version of this convergence is that, if $X$ follows the discrete distribution above, the continuous distribution above is the distribution of $X_\sigma=X+\sigma Z$ ... – Did Nov 24 '15 at 8:37
• ... with $Z$ standard normal independent of $X$, and, naturally, $X_\sigma\to X$ almost surely. – Did Nov 24 '15 at 8:40
• @zhoraster : It is my understanding that a discrete measure is a measure that has all volume concentrated in at most countably many points, as defined here. Indeed this is a linear combination of delta-measures, except that there may be infinitely many. – Josse van Dobben de Bruyn Nov 25 '15 at 9:16
• In Physics, there is a minimum distance scale (ultraviolet cutoffs) between point charges. This puts an important restriction on the discrete measures that need to be considered. – Paul Nov 26 '15 at 13:39