Mathema Tics: E. Hille

from which, dividing through by N, and letting N -a, we will get X(h) . a(h) en(h) e. Now letting e -O 0, the proof is complete. The proof of (E) reproduces the combinatorial essence of G. D. Birkhoff's proof of the Strong Ergodic Theorem.2 We rely on the fact that the "components" of h form a Boolean algebra, and may be treated like sets. 1 L. Kantorovitch. "Lineare halbgeordnete Raume," Math. Sbornik, 2, 121-68 (1937). Cf. also H. Freudenthal, "Teilweise geordnete Moduln," Proc. Akad. Wet. Amsterdam, 39, 641-51 (1936). 2 J. von Neumann, "Proof of the Quasi-Ergodic Hypothesis," these PROCEEDINGS, 18, 70-82 (1932). Our method is that used by G. D. Birkhoff for his stronger result; cf. "Proof of a Recurrence Theorem for Strongly Transitive Systems, and Proof of the Ergodic Theorem," these PROCEEDINGS, 17, 650-60 (1931). In the sense of the author's "On the combination of subalgebras," Proc. Camb. Phil. Soc., 29, 441-64 (1933). Synonyms are "Verband" (Fr. Klein) and "structure" (O. Ore). We shall use the notationf '.g for sup (f, g) andf g for inf (J, g). 4 S. Banach, "Theorie des operations lineaires," Warsaw, 1933. By general consent, the "B-spaces" of Banach, op. cit., are called Banach spaces; they are complete, metric, linear spaces. 6 In the sense of G. D. Birkhoff and Paul Smith, "Structure Analysis of Surface Transformation," Jour. Math., 7, 365 (1928). * The author is much indebted to J. von Neumann for suggesting that this lemma could be generalized. He is also indebted to S. Ulam for many conversations on the whole subject.