The Riesz representation theorem
The Riesz representation theorem provides an one-to-one correspondence between positive linear functionals on \(C_0(\Omega)\) and measures on the locally compact space \(\Omega\), viz., to each such functional \(\ell\) there exists a unique regular (Baire or Borel) measure on \(\Omega\) such that \(\ell(f)= \int_\Omega f\,d\mu\) for all \(f\in C_0(\Omega)\). The author proves this theorem by reducing it to the compact metric case and eventually to the case of the Cantor space \(\Omega_{\text{C}} = \{0,1\}^{\mathbb {N}}\), which can be handled easily since the continuous step functions are dense in \(C(\Omega_{\text{C}})\). The proof given here has some features in common with those devised by \textit{V. S. Varadarajan} [Fundam. Math. 46, 209--220 (1959; Zbl 0121.32704)], \textit{L. Báez-Duarte} [Stud. Appl. Math\ 49, 401--403 (1970; Zbl 0212.08301)] and \textit{D. J. K. Garling} [Proc. Camb. Philos. Soc. 73, 459--460 (1973; Zbl 0253.46022)].
scientific article; zbMATH DE number 5543022