# Download Calculus: Several Variables, Seventh Edition by Robert A. Adams, Christopher Essex PDF

By Robert A. Adams, Christopher Essex

Confirmed in North the USA and out of the country, this vintage textual content has earned a name for nice accuracy and mathematical rigour. prior versions were praised for delivering entire and targeted statements of theorems, utilizing geometric reasoning in utilized difficulties, and for providing quite a number functions around the sciences. Written in a transparent, coherent, and readable shape, Calculus: numerous Variables makes scholar comprehension a transparent precedence. This 7th variation encompasses a new co-author, Dr. Christopher Essex, who has been invited to give a contribution his detailed kind and method of the subject matter. teachers and scholars will relish revised routines, higher emphasis on differential equations, and new pedagogical good points.

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**Example text**

MEASURABLE FUNCTIONS 26 By Egoroff's Theorem, there exists a closed set K C E such that p(E - K) < E/2 and {SnE} converges uniformly to f on K. Note that µ((a, b) - K) < µ((a, b) - E) + µ(E - K) < E. Since each sn I K is continuous on K and the convergence is uniform, the function f l K is continuous on K. This completes the proof. 19 (LUSIN's THEOREM). If f : R -> R is a measurable function, then for each E > 0 there exist a closed set E and a continuous function g : R -> R such that p(R - E) < E and f = g on E.

If f : [a, b] -* R is Lebesgue integrable on [a, b], then for each R such that fa If - 91 < E. E > 0 there exists a continuous function g [a, b] PROOF. Suppose first that f is bounded and let M be a bound for f. Let e > 0. By Lusin's Theorem, there exist a continuous function g : [a, b] -* R and LEBESGUE INTEGRAL 47 a measurable set E C [a, b] such that g = f on [a, b] - E, ,u(E) < 612M, and g is bounded by M. We then have b Ia Now suppose that f is unbounded and let E > 0. 10, there exists an integer M such that fA IfI < E/2 where A = JxE [a, b] > M}.

If {f} converges pointwise almost everywhere on E to a function f, then for each 71 > 0 there exists a measurable set H C E such that µ(E - H) < 71 and Ifnj converges uniformly to f on H. PROOF. 7. Let B be the set of all points x E E for which {f(x)} converges to f (x). Let k be any positive integer. For each positive integer p, let BP ={xEB: Ifn(x) - f(x)l <1/kforalln>p}. Then each BP is a measurable set and 00 BP C BP+ 1 B = U BP , and P=1 lim µ (B - BP) = 0. 11 (b). Choose an integer Pk such that µ (B - BPk ) < 71/2 k This produces a sequence JBk k } of subsets of B.