Lectures on the theory of functions of real variables Volume 2

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1912 Excerpt: ...For let S, J£ be a pair of complementary e/4 normal enclosures belonging to 21; let g, F be similar enclosures of SB. Let e = Z»(g, 1?), f-Dv% F). Then e e/2, f e/2, by 355, 2. Now ® = Dv((£, 5) is a normal metric enclosure of 35. Moreover its cells g which contain points of 35 and C(35) lie among the cells of e, f. Hence ae+T+ Thus by 357, 35 is measurable. 2. Let?(, S 6e measurable. Let Then For Hence Forlet o,-at-aI, o.-a.-a,.. For uniformity let us set at = 8. Then a = 2o». As each Q„ is measurable 3 = 2 = lim(ai+ +o.)--00 = lira in. 361. Let 2Ij, S be measurable and their union 21 limited. If 3) = Dv 2l„S 0, ft 18 measurable. For let 8 lie in the metric set 2ft; let ® + D=m, an+4.=a» as usual. Now 3) denoting the points common to all the a„, no point of D can lie in all of the a„, hence it lies in some one or more of the An. Thus DAn. (1 On the other hand, a point of An lies in some Am, hence it does not lie in a„. Hence it does not lie in 3). Thus it lies in D. Hence AnD. (2 From 1), 2) we have D= A As each An is measurable, so is D. Hence 3) is. 362. If aa is an enumerable set of measurable aggregates, their divisor 3) is measurable, and © = liman. B=ae For as usual let 2, An be the complements of 3), 8, with respect to some metric set 9ft. Then 2=M»j, 4.4.+1. Hence by 360, 3-Km A.. Let era=(emil, em2, e,„,8...) + em, (3 then em is measurable. By this process the metric or measurable cell em falls into an enumerable set of non-overlapping measurable cells, as indicated in 3). If we suppose this decomposition to take place for each cell of ©, we shall say we have superimposed 8 on S. 364. (TF. H. Young. Let S be any complete set in limited SJ. Then = Max S. (1 For le...