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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
T h e p r i n t a b l e v e r s i o n i s n o l o n g e r s u p p o r t e d a n d m a y h a v e r e n d e r i n g e r r o r s . P l e a s e u p d a t e y o u r b r o w s e r b o o k m a r k s a n d p l e a s e u s e t h e d e f a u l t b r o w s e r p r i n t f u n c t i o n i n s t e a d .
Suppose that
(
X
,
B
,
μ
)
{\displaystyle (X,{\mathcal {B}},\mu )}
is a probability space , that
T
:
X
→
X
{\displaystyle T:X\to X}
is a (possibly noninvertible) measure-preserving transformation , and that
f
∈
L
1
(
μ
,
R
)
{\displaystyle f\in L^{1}(\mu ,\mathbb {R} )}
. Define
f
∗
{\displaystyle f^{*}}
by
f
∗
=
sup
N
≥
1
1
N
∑
i
=
0
N
−
1
f
∘
T
i
.
{\displaystyle f^{*}=\sup _{N\geq 1}{\frac {1}{N}}\sum _{i=0}^{N-1}f\circ T^{i}.}
Then the maximal ergodic theorem states that
∫
f
∗
>
λ
f
d
μ
≥
λ
⋅
μ
{
f
∗
>
λ
}
{\displaystyle \int _{f^{*}>\lambda }f\,d\mu \geq \lambda \cdot \mu \{f^{*}>\lambda \}}
for any λ ∈ R .
This theorem is used to prove the point-wise ergodic theorem .
References
Keane, Michael; Petersen, Karl (2006), "Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem", Dynamics & Stochastics , Institute of Mathematical Statistics Lecture Notes - Monograph Series, vol. 48, pp. 248–251, arXiv :math/0004070 , doi :10.1214/074921706000000266 , ISBN 0-940600-64-1 .
t
e
t
e
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Maximal_ergodic_theorem&oldid=1213448147 "
C a t e g o r i e s :
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H i d d e n c a t e g o r i e s :
● A r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s f r o m M a r c h 2 0 2 4
● A l l a r t i c l e s n e e d i n g a d d i t i o n a l r e f e r e n c e s
● A l l s t u b a r t i c l e s
● T h i s p a g e w a s l a s t e d i t e d o n 1 3 M a r c h 2 0 2 4 , a t 0 3 : 3 7 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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