Riesz–Markov–Kakutani representation theorem

Remarks

Clauses 2 to 4 say that $\mu$ is a Radon measure. The proof sets $\mu(U) = \sup \set{\Lambda f : 0 \le f \le 1, \ \operatorname{supp} f \subseteq U}$ for open $U$, extends it to an outer measure, and checks that the Borel sets are measurable. Urysohn's lemma for locally compact Hausdorff spaces supplies the functions $f$ that the construction needs.

Example: on $X = \R$, the Riemann integral $\Lambda f = \int_{-\infty}^{\infty} f(x) \, dx$ is positive, and the measure it gives is Lebesgue measure. The evaluation $\Lambda f = f(0)$ gives the Dirac measure at $0$.

Let $X$ be compact Hausdorff and give $C(X)$ the sup norm. A related form identifies the dual of $C(X)$ with the space of regular signed Borel measures on $X$, normed by total variation.

Going deeper

For a student

The theorem is the bridge between functional analysis and measure theory. It identifies the dual of $C(X)$ for compact $X$ with a space of measures, and so gives the weak-* topology on measures that underlies weak convergence of probability measures. Haar measure on a locally compact group is often built by constructing an invariant positive functional and then applying this theorem.