Nonstandard proof of spectral theorem (Q2730971)

The author proves the existence of the spectral measure of a bounded selfadjoint operator by methods of nonstandard analysis. The idea to use a hyperfinite extension of the given operator goes back to \textit{L. R. Moore jun.} [Trans. Am. Math. Soc. 218, 285-295 (1976; Zbl 0341.47018)] who proved already the existence of a spectral resolution of the identity for such an operator in exact the same manner as is done in the present paper. The author seems to do not know Moore's result. What is really new here in this paper is the extension of the spectral resolution to a spectral measure by using Loeb measure theory. On the other hand the proof is not quite correct since the projections \(G (A)\) need not leave invariant the external space of all standard elements, so one cannot use Theorem 3.2 as the author did. What should be used instead is the relative compactness of bounded sets of operators with respect to the weak operator topology or a nonstandard version of this fact.