Inmathematics, theFreudenthal spectral theorem is a result inRiesz space theory proved byHans Freudenthal in 1936. It roughly states that any element dominated by a positive element in aRiesz space with theprincipal projection property can in a sense be approximated uniformly bysimple functions.

Numerous well-known results may be derived from the Freudenthal spectral theorem. The well-knownRadon–Nikodym theorem, the validity of thePoisson formula and thespectral theorem from the theory ofnormal operators can all be shown to follow as special cases of the Freudenthal spectral theorem.

Lete be any positive element in a Riesz spaceE. A positive element ofp inE is called a component ofe if${\displaystyle p\wedge (e-p)=0}$. If${\displaystyle p_1,p_2,\dots ,p_n}$ are pairwisedisjoint components ofe, any real linear combination of${\displaystyle p_1,p_2,\dots ,p_n}$ is called ane-simple function.

The Freudenthal spectral theorem states: LetE be any Riesz space with the principal projection property ande any positive element inE. Then for any elementf in the principal ideal generated bye, there exist sequences${\displaystyle \{s_n\}}$ and${\displaystyle \{t_n\}}$ ofe-simple functions, such that${\displaystyle \{s_n\}}$ is monotone increasing and convergese-uniformly tof, and${\displaystyle \{t_n\}}$ is monotone decreasing and convergese-uniformly tof.

Let${\displaystyle (X,\mathrm{\Sigma })}$ be ameasure space and${\displaystyle M_{\sigma }}$ the real space ofsigned${\displaystyle \sigma }$-additive measures on${\displaystyle (X,\mathrm{\Sigma })}$. It can be shown that${\displaystyle M_{\sigma }}$ is aDedekind completeBanach Lattice with thetotal variation norm, and hence has theprincipal projection property. For any positive measure${\displaystyle \mu }$,${\displaystyle \mu }$-simple functions (as defined above) can be shown to correspond exactly to${\displaystyle \mu }$-measurablesimple functions on${\displaystyle (X,\mathrm{\Sigma })}$ (in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure${\displaystyle \nu }$ in theband generated by${\displaystyle \mu }$ can be monotonously approximated from below by${\displaystyle \mu }$-measurable simple functions on${\displaystyle (X,\mathrm{\Sigma })}$, byLebesgue's monotone convergence theorem${\displaystyle \nu }$ can be shown to correspond to an${\displaystyle L^1(X,\mathrm{\Sigma },\mu )}$ function and establishes an isometric lattice isomorphism between the band generated by${\displaystyle \mu }$ and the Banach Lattice${\displaystyle L^1(X,\mathrm{\Sigma },\mu )}$.

• Radon–Nikodym theorem

• Zaanen, Adriaan C. (1996),Introduction to Operator Theory in Riesz spaces,Springer,ISBN 3-540-61989-5
• Zaanen, Adriaan C.; Luxemburg, W. A. J. (1971),Riesz spaces I,North-Holland,ISBN 0-7204-2451-8

• Theorems in functional analysis