Let 1 < p < ∞ and let Lp be the usual Banach Space of complex valued functions on a σ-finite measure space. Let (Tn), n ≧ 1, be a sequence of positive linear contractions on Lp. Hence
and
, where
is the part of Lp that consists of non-negative Lp functions. The adjoint of Tn is denoted by
which is a positive linear contraction of Lq with q = p/(p — 1).
Our purpose in this paper is to show that the alternating sequences associated with (Tn), as introduced in [2], converge almost everywhere. Complete definitions will be given later. When applied to a non negative function, however, this result is reduced to the following theorem.
(1.1) THEOREM. If (Tn) is a sequence of positive contractions of Lp then (1.2) ![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190725034638795-0231:S0008414X00000262:S0008414X00000262_inline5.gif?pub-status=live)
exists a.e. for all
.