The monotone class theorem says that the smallest monotone class containing an algebra of sets equals the sigma-algebra generated by that algebra. It extends identities proved on a generating algebra to the full sigma-algebra.
A vector space of bounded functions containing constants and indicators of a generating pi-system, and closed under uniformly bounded monotone pointwise limits, contains all bounded functions measurable for the generated sigma-algebra. This function version extends identities established on elementary integrands to every bounded measurable integrand.
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The Monotone Class Theorem is an important result in measure theory, particularly in the theory of σ-algebras and the construction of measures. It provides a way to extend certain types of sets (often related to a σ-algebra) under specific conditions. The theorem is usually stated in terms of the construction of σ-algebras from collections of sets.