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.
