Suppose that τ is a measure-preserving transformation of a measure space S with finite measure. If f is a real-valued integrable function on S then the Wiener–Wintner theorem states that there is a measure 0 set E such that the average
exists for all real λ and for all P not in E.
The special case for λ = 0 is essentially the Birkhoff ergodic theorem, from which the existence of a suitable measure 0 set E for any fixed λ, or any countable set of values λ, immediately follows. The point of the Wiener–Wintner theorem is that one can choose the measure 0 exceptional set E to be independent of λ.
This theorem was even much more generalized by the Return Times Theorem.
- Assani, I. (2001), "W/w130110", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Wiener, Norbert; Wintner, Aurel (1941), "Harmonic analysis and ergodic theory", American Journal of Mathematics 63: 415–426, doi:10.2307/2371534, ISSN 0002-9327, JSTOR 2371534, MR 0004098