Abstract
In this article, we prove two forms of conditional mean ergodic theorem for a strongly continuous semigroup of random isometric linear operators generated by a semigroup of measure-preserving measurable isomorphisms, one of which generalizes and improves several known important results.
Highlights
Introduction and the main resultsThe notion of a random normed module, which was first introduced in [ ] and subsequently elaborated in [ ], is a random generalization of that of a normed space
It is well known that the (ε, λ)-topology induced by the L -norm on an RN module is exactly the topology of convergence in probability P
It is Mustari and Taylor that earlier observed the essence of the (ε, λ)-topology, studied probability theory in Banach spaces and did many excellent works [, ] under the framework of the special RN module L (E, X), where L (E, X) is the RN module of equivalence classes of X-valued random variables defined on a probability space (, E, P), see [ ] or Section for the construction of L (E, X)
Summary
Introduction and the main resultsThe notion of a random normed module (briefly, an RN module), which was first introduced in [ ] and subsequently elaborated in [ ], is a random generalization of that of a normed space. Based on these and motivated by the idea of [ , ], the purpose of this article is to investigate the conditional mean ergodicity for a special semigroup of random linear operator on the RN module LpF (E, X) and the construction of LpF (E, X) is detailed as follows.
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