Freudenthal’s Spectral Theorem

Abstract

Let μ be a σ-finite measure in the (non-empty) point set X. As well-known, any finite linear combination of characteristic functions of measurable subsets of X is called a μ-step function. It is not difficult to see that for any 0 ≤ f ∈ L _∞ (X, μ) there exists a sequence (s _n : n = 1, 2,…) of μ-step functions such that 0 ≤ s _n ↑ f holds uniformly. A similar result holds in an Archimedean Riesz space possessing a strong unit. This result, due to H.Freudenthal (1936), is known as Freudenthal’s spectral theorem. Some preliminary theorems about projection bands will be useful. Recall that the band B in the Archimedean Riesz space E is called a projection band if E = B ⊕B^d (see section 11). In this case B^d is likewise a projection band (we have B = B^dd because E is Archimedean, so E = B^d ⊕ B^dd). As in Theorem 11.4, the band projection on the projection band B is sometimes denoted by P _B . It is obvious that${P_{{B^d}}} = I - {P_B}$,where I is the identity operator in E (i.e., I f = f for each f ∈ E, and so${P_{{B^d}}}f = f - {P_B}f$). The following holds now.