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a-algebra a-finite a.e. x G absolutely continuous Borel measurable bounded function called choose Clearly continuous function convergence in measure corollary countably additive cr-finite defined definition denote differentiable dominated convergence theorem example Exercise fdfi fi(A fi(E fi(X finite measure finitely additive fn(x following hold function f G Rn given Hence implies increasing sequence integrable functions Lebesgue integral Lebesgue measure lemma length function Let f G Let fn}n>i linear measurable function measure on X,S measure space monotone class monotone convergence theorem monotonically increasing n}n>i converges nonempty set nonnegative measurable function null set oo oo open intervals open set outer measure pairwise disjoint partition proposition Prove the following Radon-Nikodym theorem real numbers required claim Riemann integrable semi-algebra set function Show that f signed measure simple function simple measurable functions sn(x subspace uniformly Xi-i
Page xiii - Mathematics presented as a closed, linearly ordered, system of truths without reference to origin and purpose has its charm and satisfies a philosophical need. But the attitude of introverted science is unsuitable for students who seek intellectual independence rather than indoctrination; disregard for applications and intuition leads to isolation and atrophy of mathematics.