## Introduction to Measure Theory |

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absolutely continuous apply Borel set Borel-measurable bounded Card Chapter choose Clearly closed complete condition consider contains continuous function convergence convex Corollary countable decomposition defined Definition denote disjoint equality ess sup Example Exercise exists extended f and g f dx finite function f given gives the result hence holds implies inequality infinite integrable function Lebesgue Let f lim fn limit mean measurable function measurable set measure space measure zero monotone increasing notation o-algebra o-finite obtain obvious open intervals open set partition points positive Proof prove respect result follows Riemann integrable sequence sequence of measurable Show side signed measure Similarly Solution space subsets sufficient sup f suppose Theorem 12 Theorem 9 uniformly union unique values write