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**Extra resources for Calculus Equations & Answers**

**Sample text**

So f is measurable. Suppose on the contrary that /z — g > 0 on a set of positive measure. Then (Problem 10) there is /? > 0 so that h — g > p on a set A of positive measure. For all n and all points of A, fn-(Pn>h-g>p. If Eni intersects A, then Mni — ^ni > P; these sets cover A and hence have aggregate measure at least /x(A). Thus for each w, U(f Pn) - L(f Pn) = E ( ^ - - ^ - ) M ( £ - ) > P/^(^)This contradicts our assumption that U( f Pn) — L( f Pn) —> 0. ill Problem 10. Show that if ^ is a measurable function and {x : k(x) > 0} has positive measure, then {x : k(x) > p} has positive measure for some p > 0.

If / is of the form (0,fc)or (a, 1), the same argument works by considering the single complementary interval, ill Problem 1. Carry out the proof of Proposition 3 in the case } =(0,b). II Our definition says £ is measurable if £ splits (0, 1) additively. Proposition 3 shows that £ is measurable only if £ splits every subinterval of (0, 1) additively. We next show that £ is measurable if and only if £ splits every subset T additively. This result is due to Caratheodory and has become the modern definition of measurability in all general settings.

Hint: See the proof of Proposition 6, Chapter 5. ""HI Now we show that the general integral is a limit of Riemann sums, and prove the linearity properties. f{Ci)iJi{Ei)^ where P = {£J is a partition of S, and Q e Ei for each /. We will use the notation R(f, P, c) for such a sum only if P is an admissible partition of S. Riemann sums are now generally infinite series, and the admissibility condition ensures that the series converge absolutely. The pairs (P, c) are directed as usual, with (P, c) >- (Q, cO if P > Q^ and with this agreement {R(f, P, c)} is a net on the pairs (P, c) with P admissible.