Freudenthal spectral theorem

Statement

Let e be any positive element in a Riesz space E. A positive element of p in E is called a component of e if p\wedge(e-p)=0. If p_1,p_2,\ldots,p_n are pairwise disjoint components of e, any real linear combination of p_1,p_2,\ldots,p_n is called an e-simple function.

The Freudenthal spectral theorem states: Let E be any Riesz space with the principal projection property and e any positive element in E. Then for any element f in the principal ideal generated by e, there exist sequences {s_n} and {t_n} of e-simple functions, such that {s_n} is monotone increasing and converges e-uniformly to f, and {t_n} is monotone decreasing and converges e-uniformly to f.

Relation to the Radon–Nikodym theorem

Let (X,\Sigma) be a measure space and M_\sigma the real space of signed \sigma-additive measures on (X,\Sigma). It can be shown that M_\sigma is a Dedekind complete Banach Lattice with the total variation norm, and hence has the principal projection property. For any positive measure \mu, \mu-simple functions (as defined above) can be shown to correspond exactly to \mu-measurable simple functions on (X,\Sigma) (in the usual sense). Moreover, since by the Freudenthal spectral theorem, any measure \nu in the band generated by \mu can be monotonously approximated from below by \mu-measurable simple functions on (X,\Sigma), by Lebesgue's monotone convergence theorem \nu can be shown to correspond to an L^1(X,\Sigma,\mu) function and establishes an isometric lattice isomorphism between the band generated by \mu and the Banach Lattice L^1(X,\Sigma,\mu).

References

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