1. Introduction. Let (X,
, m) be a σ-finite measure space and let T be a positive linear operator on L1 = L1(X,
, m). T is called Markovian if
(1.1)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00015182/resource/name/S0008414X00015182_eqn1.gif?pub-status=live)
T is called sub-Markovian if
(1.2)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00015182/resource/name/S0008414X00015182_eqn2.gif?pub-status=live)
All sets and functions are assumed measurable; all relations and statements are assumed to hold modulo sets of measure zero.
For a sequence of L1+ functions (ƒ0, ƒ1, ƒ2, …), let
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00015182/resource/name/S0008414X00015182_eqn3.gif?pub-status=live)
(ƒn) is called a super additive sequence or process, and (sn) a super additive sum relative to a positive linear operator T on L1 if
(1.3)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00015182/resource/name/S0008414X00015182_eqn4.gif?pub-status=live)
and
(1.4)![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00015182/resource/name/S0008414X00015182_eqn5.gif?pub-status=live)