Birkhoff’s ergodic theorem gives us information about the convergence of sequences of functions. We are then interested in studying the mean and pointwise convergence of these sequences, but in the case the sequence is a strictly increasing sequence of positive integers. With that goal in mind, we will define uniform sequences and study the pointwise convergence for these sequences. We will also explore the possibility that there exists some sequences for which the convergence of the sequence does not occur. We will present a result of Alexandra Bellow that says that such sequences exist. Finally, we will prove a result which establishes an equivalence between the notion of a strongly mixing transformation and the convergence of a sequence that uses “weights” which satisfies certain properties.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.