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Ito's Theorem and the Ratio Theorem

Ito's Theorem and the Ratio Theorem F. J. YEADON Let (X, Z, /i) be a ^-finite measure space, and let T be a linear positive 1 1 contraction of L = I}(X, Z, n) into itself; that is if / e L and / ^ 0, then 7/^ 0 j n- l and JT/ ^ J/ . We denote by T the averages defined by T f = - £ T f. n n k = o Ito's theorem ([5], Theorem 1) may be stated in the form: If for some p e I}(X, Z, /i) with p(x) > Ofor almost all xe X the set of averages {T p} 1 1 is weakly relatively compact in L , then for each /eL ^,!,^ ) the averages T f converge fi-almost everywhere and in I}-norm. Ito's proof and the shorter version due to Kim [6] make use of the Chacon- Ornstein ratio theorem which states that n - 1 In-I k k if f,gel} and g ^ 0 almost everywhere in X, then the ratios £ T f £ T g fc = converge almost everywhere on the set where ]£ T g > 0. k = 0 In this paper we provide a proof http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Ito's Theorem and the Ratio Theorem

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/12.4.279
Publisher site
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Abstract

F. J. YEADON Let (X, Z, /i) be a ^-finite measure space, and let T be a linear positive 1 1 contraction of L = I}(X, Z, n) into itself; that is if / e L and / ^ 0, then 7/^ 0 j n- l and JT/ ^ J/ . We denote by T the averages defined by T f = - £ T f. n n k = o Ito's theorem ([5], Theorem 1) may be stated in the form: If for some p e I}(X, Z, /i) with p(x) > Ofor almost all xe X the set of averages {T p} 1 1 is weakly relatively compact in L , then for each /eL ^,!,^ ) the averages T f converge fi-almost everywhere and in I}-norm. Ito's proof and the shorter version due to Kim [6] make use of the Chacon- Ornstein ratio theorem which states that n - 1 In-I k k if f,gel} and g ^ 0 almost everywhere in X, then the ratios £ T f £ T g fc = converge almost everywhere on the set where ]£ T g > 0. k = 0 In this paper we provide a proof

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Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1980

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