Rice University Studies Volume 7, No. 4

Rice University Studies Volume 7, No. 4

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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. 1920 edition. Excerpt: ...measurable in the Borel sense, and have integrals. The equation (5) is such a property, also the following ones.f (6') If qj(M) constitute an increasing sequence of functions, For those properties which are essential merely for the study of Stieltjes potentials, see l0). For a more general and systematic study, see 7). f A different method of extending the field of definition of the integral is suggested by Lebesgue. u) H. Lebesgue, " Sur l'integrale de Stieltjes et sur les operations fonctionelles Uneaires," Comptes Rendus h. de l'Acad. des Sciences, Vol. 150 (1910) pp. 86-88. qi(M)df(e)+b qt(M)df(e). In particular, the property (6') provides a definition for the integral for functions q(M) that become infinite at certain points M. 1.3. It will be necessary to consider iterated integrals, and functions q(Mu M) depending on two point arguments. If q(Mi, M) is continuous in both point arguments we have such identities as the following: l8) In fact, these identities follow immediately from the approximation formula (5"). By means of (6') and (6'") the extension to bounded functions measurable in the Borel sense is also immediate. If the function q(Mu M) is not bounded we may write q (Mu M)-I q(M- M), if q(Mu M) n. = n, otherwise, and in order for (7) to hold it is sufficient that ") W. H. Young, " Integration with respect to a function of bounded variation," Proceedings London Mathematical Society, Vol. 13 (1914), pp. 97-150; see page 149. where t/ and tg denote the total variation functions of f(e) and g(?) respectively, remain finite; for (7') to hold it is sufficient that remains finite as n becomes infinite. In fact, since the righthand member remains finite, as n becomes infinite, it follows that lim...
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  • Paperback | 28 pages
  • 189 x 246 x 2mm | 68g
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236573625
  • 9781236573629