Monotone class

A monotone class of sets is closed under increasing countable unions and decreasing countable intersections.

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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