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: ... enclosure of 21. Thus I 2, 2n + £) = 2?„ + e, which establishes 1). 357. Let @ be a distinct measurable enclosure of 21. Let f denote those cells containing points of the complement A. If for each e 0 there exists an g such that J e, then 21 is measurable. For let (g = e + f. Then c 21. Hence c 21 by 348. But ii=t+fa+«. Hence %-%«, and thus IF--w 358. 1. The divisor D of two measurable sets 21, S3 t also measurable. For let (g, E be a pair of complementary c/4 normal enclosures belonging to 21; let g, F be similar enclosures of 83. Let e = Zi(£, J?), f = ZK& &) Then e e/2, f e/2, by 355, 2. Now ® = Dv(@, 5) is a normal metric enclosure of 35. Moreover its cells 9 which contain points of 35 and C(35) lie among the cells of e, f. Hence Thus bv 357, 35 is measurable. 2. Let 21, 58 be measurable. Let Then For Hence S3 is measurable by 353, 1, and the relation 1) holds by the same theorem. 360. Let fl1fl%---bea set of measurable aggregates whose union 21 is limited. Then 21 is measurable, and a=uma,. 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 6, we shall say we have superimposed & on (g. 364. (W. H. Young. Let 6 be any complete set in limited 91. Then a = Max 6. (1 For let 81 lie within a cube 9K, and let A =. 9W-a, C= 2K-6 be as usual the complementary sets. Let S= b„ be a border set of S 328. It is also a nonoverlapping enclosure of 0; we may suppose it is a standard enclosure of C. Let E be a standard e-enclosure of A. Let us superimpose E on SB, getting a measurable enclosure A of both (7 »nd A. Then C=7AAA. Hpncs 6 = aw _ c= s&...