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Section CHAPTER I MEASURES 1 Rings and rings
The lemma on monotone classes
Set functions measures
30 other sections not shown
a.e. PROOF absolute continuity AC with respect assume Borel set bounded complement fn(x convergence in measure COROLLARY countably additive definition disjoint union dominated convergence E U F exists a measurable extended real numbers extended real valued f and g f dp f is integrable f is measurable f U g Fatou's lemma finite measure space finite signed measure fm(x fn dp formula Fubini's theorem function f g are measurable g dp gn(x hence f I(fn integrable function integrable with respect integral of f iterated integral Jf dp Let f liminf limsup locally measurable set measurable function monotone class monotone convergence theorem moreover mutually disjoint notation null set outer measure pointwise Radon-Nikodym theorem real valued functions Riesz-Fischer theorem ring sequence f sequence of measurable sequence of sets set function Similarly simple functions Suppose f T-finite uniform convergence valued functions defined