A Tour of the Calculus by David Berlinski

By David Berlinski

In its biggest point, the calculus services as a celestial measuring tape, in a position to order the endless expanse of the universe. Time and area are given names, issues, and boundaries; doubtless intractable difficulties of movement, progress, and shape are lowered to answerable questions. Calculus was once humanity's first try to signify the area and maybe its maximum meditation at the subject matter of continuity. Charts and graphs all through.

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Example text

T( J ) 5 CF'Ll C( I k ) . Let J = [a,61 c 11 U . - . U In+l, and assume no n of these intervals cover J . If any 1k is disjoint from J we are done. Let us assume, to be definite, that 1,+1 = (c, d ) with a < c < d < b. Let J 1 = [a,c] and J 2 = [ d ,61 be the two subintervals of J not covered by 1,+1. No interval I k , k = 1,. . , n, can intersect both J 1 and 1 2 , for such an interval would cover In+l, and n of the Ik 24 A PRIMER OF LEBESGUE INTEGRATION would cover J. Therefore some of the intervals 11, .

Note that subsets of sets of measure zero have measure zero. 1111111 Since m(E) is to be a generalization of length, we need to know that m ( I ) = C(1) for every interval I c ( 0 , l ) . That is the content of the next proposition and problem. Proposition 2. I f J = [a,b] c (0, l),or if J = ( a , 6) c (0, l), then m ( J )= C ( J ) . Proof. Clearly m ( J )5 C ( J ) since { J } is a one-interval covering of J of total length C ( J ) . We show by induction that if 11, . . , I, is any finite covering of J by intervals, then C ( J ) 5 C(Ik).

The verification of these facts is the program for this chapter. A cautionary word about notation and nomenclature: most texts use m* for our function m and refer to it as Lebesgue outer measure. The unadorned letter m is used by these authors for the restriction of m* to the measurable sets, and this restricted function is called Lebesgue measure. We will stick to m, defined on all subsets of (0, l),and call m(E)the measure of E whether E is measurable or not. , after this chapter) we only consider m(E) for measurable sets since measurability is essential for m to have the critical property of additivity.

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