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I will be doing a short exposition this week on the Stone-Weierstrass for a course in General Topology and Measure Theory. I am planning on going through the proof of the theorem, the proof of the complex version and then I will state its version for vector lattices of continuous functions. Then I will move on to applications. Up to today, these are the ones I have:

  1. Particular case on dimension 1: Weierstrass approximation theorem;
  2. Trigonometric polynomials on $\mathbb{R}$ are dense in the set of periodic functions.

Then I would like to have some more examples more closely related to topology/measure. As far as topology is concerned, I have found two possibilities:

  1. If $X$ is a compact metric space, then $C(X)$ is separable;

  2. Locally compact version of Stone-Weierstrass.

However, I did not find an example of application in measure theory.

So I would be very appreciated if anyone could tell me about one such example and, of course, I would be very happy to see more examples of applications of the theorem!

  • You could say that trigonometric polynomials being dense in periodic functions could motivate Fourier expansions and coefficients. In a similar manner, Legendre polynomials in $[-1,1]$ might motivate Gaussian Quadrature (a near hocus-pocus way to calculate integrals, which is measure-theoretic?). –  Nov 27 '17 at 19:22
  • See this question e.g. – Henno Brandsma Nov 27 '17 at 21:35

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