Time average of an observable

For an observable $f$ and a probability-preserving transformation $T$ , its time average over $N\geq1$ steps is

A_Nf(x)=\frac1N\sum_{n=0}^{N-1}f(T^nx).

Compare this orbit sample with the space average under $\mu$ .

Visits to events

For $f=1_E$ , the time average is the fraction of the first $N$ iterates lying in $E$ . The pointwise ergodic theorem identifies its almost-everywhere limit. In a nonergodic system, that limit can depend on the starting point.