Birkhoff,s Ergodic Theorem
المؤلف:
Cornfeld, I.; Fomin, S.; and Sinai, Ya. G
المصدر:
Appendix 3 in Ergodic Theory. New York: Springer-Verlag, 1982.
الجزء والصفحة:
...
6-10-2021
1699
Birkhoff's Ergodic Theorem
Let 
be an ergodic endomorphism of the probability space 
and let 
be a real-valued measurable function. Then for almost every 
, we have
 |
(1)
|
as 
. To illustrate this, take 
to be the characteristic function of some subset 
of 
so that
![f(x)=<span style=]() {1 if x in A; 0 if x not in A. " src="https://mathworld.wolfram.com/images/equations/BirkhoffsErgodicTheorem/NumberedEquation2.gif" style="height:41px; width:125px" /> |
(2)
|
The left-hand side of (1) just says how often the orbit of 
(that is, the points 
, 
, 
, ...) lies in 
, and the right-hand side is just the measure of 
. Thus, for an ergodic endomorphism, "space-averages = time-averages almost everywhere." Moreover, if 
is continuous and uniquely ergodic with Borel measure 
and 
is continuous, then we can replace the almost everywhere convergence in (1) with "everywhere."
REFERENCES:
Cornfeld, I.; Fomin, S.; and Sinai, Ya. G. Appendix 3 in Ergodic Theory. New York: Springer-Verlag, 1982.
الاكثر قراءة في الرياضيات التطبيقية
اخر الاخبار
اخبار العتبة العباسية المقدسة
