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''in the [[Phase space|phase space]]  $  X $(
a metrizable compactum) of a [[Topological dynamical system|topological dynamical system]]  $  \{ S _ {t} \} $(
a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] or a [[Cascade|cascade]]) corresponding to a normalized ergodic [[Invariant measure|invariant measure]]  $  \mu $''

A set of points  $  x \in X $
such that:

a) for every continuous function  $  f :  X \rightarrow \mathbf R $
the  "time average" 

$$ 

\frac{1}{T}
 \int\limits _ { 0 } ^ { T }  f ( S _ {t} x )  d t
 \rightarrow  \int\limits _ { X } f  d \mu \  \textrm{ as }  T \rightarrow \infty ;
$$

b)  $  \mu ( U) > 0 $
for every neighbourhood  $  U $
of  $  x $.

A point for which the limit of the time average in a) exists for every continuous  $  f $
is called quasi-regular. For such a point this limit has the form  $  \int f  d \mu $, 
where  $  \mu $
is some normalized invariant measure, depending on  $  x $
and not necessarily ergodic. If b) holds for this  $  \mu $, 
then the point is called a density point, if this  $  \mu $
is ergodic, then the point is called transitive; when both conditions hold, the point is called regular. The set of non-regular points has measure zero relative to any normalized invariant measure. The partition of the set of regular points into ergodic sets corresponding to different  $  \mu $
is the strongest realization of the idea of decomposing a dynamical system into ergodic components (which is valid under stronger restrictions on the system).

Ergodic sets were introduced by N.N. Bogolyubov and N.M. Krylov [[#References|[1]]]. For other accounts, discussions of various generalizations and related questions see the references to [[Invariant measure|Invariant measure]] 1) and [[Metric transitivity|Metric transitivity]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Krylov,   N. [N.N. Bogolyubov] Bogoliouboff,   "La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non-linéare"  ''Ann. of Math. Ser. (2)'' , '''38'''  (1937)  pp. 65–113</TD></TR></table>

====Comments====
A good account of ergodic sets is given in [[#References|[a1]]]. Closely related is the notion of a generic point (with respect to a normalized invariant measure  $  \mu $): 
A quasi-regular point such that for every continuous  $  f :  X \rightarrow \mathbf R $
the limit in a) is  $  \int f  d \mu $, 
where  $  \mu $
is the given measure.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.C. Oxtoby,   "Ergodic sets"  ''Bull. Amer. Math. Soc.'' , '''58'''  (1952)  pp. 116–136</TD></TR></table>