Abstract
Let (X, S, μ) and (Y, T, ν) be two measure spaces,K(g)=∫ Y k(x,y)g(y)dv(y) ξ +=max (ξ,0), and\(\delta (K) = \sup _{x_1 ,x_2 \in X} \int {{}_Y(k(x_1 } y) - (k(x_2 ,y))^ + dv(y)\). Two integral inequalities (the first of which has a simple geometrical interpretation) involvingδ(K) are proved. Generalizations of theorems about infinite stochastic matrices and convergence of superpositions of sequences of integral operators are obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. L. Dobrushin,Central limit theorem for non stationary Markov chains I, II, Theory Prob. Applications, I (1956), 65–80 and 329–383 (English translation).
J. Hajnal,Weak ergodicity in nonhomogeneous Markov chains, Proc. Cambridge Philos. Soc.54 (1958), 233–246.
A. Paz,Ergodic theorems for infinite probabilistic tables, Ann. Math. Statistics41 (1970), 539–550.
A. Paz and M. Reichaw,Ergodic theorems for sequences of infinite stochastic matrices, Proc. Cambridge Philos. Soc.63 (1967), 777–784.
M. Reichaw,On the convergence of superpositions of a sequence of operators, Studia Math.25 (1965), 343–351.
M. Reichaw,Supplement to my paper “On the convergence of superpositions of a sequence of operators”, Studia Math.28 (1966/7), 361.
Author information
Authors and Affiliations
Additional information
Dedicated to the memory of E. Jabotinsky
An erratum to this article is available at http://dx.doi.org/10.1007/BF02771578.
Rights and permissions
About this article
Cite this article
Blum, J.R., Reichaw, M. Two integral inequalities. Israel J. Math. 9, 20–26 (1971). https://doi.org/10.1007/BF02771615
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02771615