A multiple ergodic average is an average of products of functions evaluated at different iterates of a measure-preserving
map 
.
For bounded measurable functions 
, ..., 
and integer slopes 
, ..., 
,
a linear multiple ergodic average is
 |
Negative slopes require an invertible map whose inverse is a measurable function. For one function, Birkhoff's ergodic theorem describes the almost everywhere convergence of these averages under its integrability and ergodicity hypotheses.
