Soe Classical Ergodic Theores Matt Rosenzweig Contents Classical Ergodic Theores. Mean Ergodic Theores........................................2 Maxial Ergodic Theore..................................... 3.3 Pointwise Ergodic Theore..................................... 5.4 Application to Law of Large ubers............................... 6 Classical Ergodic Theores. Mean Ergodic Theores Proposition. (von euann Ergodic Theore) Let H be a Hilbert space, T : H H be a unitary operator. Then for all f H, the it f : A nf exists in H. In fact, f P T f, where P T is the orthogonal projection onto H T : f H : T f f. Proof. We first reark that H T is evidently closed by the continuity of the T. Consider two cases. If f H T, then A f f for all, so the stateent is obvious. Suppose f / H T. Consider f T g g for soe g H. Then A f ( T + T ) g 0, since T is bounded. Set H diff T g g : g H. I clai that A f 0 for all f H diff. oting that A for all and applying the triangle inequality,for f H diff and a sequence T g n g n f, A f A f A (T g n g n ) + A (T g n g n ) f (T g n g n ) + A (T g n g n ) f (T g n g n ) + 2 g n 0, n, The proof will be coplete if we can show that H H T H diff, for which it suffices to show that H T H diff. Suppose T f f, g H. Then f, T g g T f, T g f, g f, g f, g 0, where we use the fact that T is unitary. By continuity, we obtain f H diff. For the reverse inclusion, if f H diff, then 0 f, T f f f, T f T f, T f T f f, T f Taking the coplex conjugate of the RHS and adding it to f, T f f gives 0 T f f, T f f f H T Definition 2. A easure-preserving syste is (, A, µ, T ) consists of a probability space (, A, µ) and a easurable transforation T : such that µ(t (E)) µ(e) for all E F. We define a function T f : f T.
Lea 3. For p, the operator T : L p L p, f T f is an isoetry. Moreover, T : L 2 L 2 is unitary. Proof. Since the siple functions are dense in L p for all p [, ], it suffices to verify the clai on this subspace. The case p is evident, so we assue p <. Let f n a i Ei be a siple function where we ay assue that the E i are pairwise disjoint. The preiages T (E j ) are pairwise disjoint. Since T is easure-preserving, p T f P dµ a i T (E i) dµ a i p µ(t (E j )) a i p µ(e i )dµ f p dµ For the second assertion, we need only verify that T ft gdµ fgdµ for all f, g L2 (, µ). By density, we ay assue that f n a i Ei, g j b j Fj, where the E i, F j are each pairwise disjoint collections. ( n T ft gdµ a i T (E i)) b j T (E j) dµ j j j j j a i b j T (E i) T (F j)dµ a i b j µ(t (E i F j )) a i b j µ(e i F j ) ( n fgdµ a i Ei ) b j Fj dµ j Proposition 4. (Mean Ergodic Theore) Let (, F, µ, T ) be a easure-preserving syste, p <, and f L p. Then A f converges in L p. Proof. Since T is an isoetry L p L p, p <, it follows that A L p L. It is enough to prove p the stateent for a dense subspace of all L p, in particular L. Choose f L. Von euann s ergodic theore tells us that A f f L in L 2 -nor. By Hölder s inequality, A f f in L p -nor, for p 2. For p > 2, observe that A f f p dµ A f f p 2 A f f 2 dµ (2 f L ) p 2 A f f 2 dµ 0, Definition 5. Let (, A, µ) be a probability space and T :. We say that T is ergodic if for all E F, T (E) E a.e. µ(e) 0 or µ(e). Lea 6. T is ergodic if and only if for all f L, T f f a.e. f is constant a.e. Proof. We first show the direction. Let E A satisfy T (E) E a.e.. Then E L and T ( E ) T (E) E a.e. E is constant a.e. For, we assue that f is real-valued (for the coplex-valued case, just decopose f into its real and iaginary parts). Define E a : f a Then E a satisfies T (E a ) E a a.e. µ(e a ) 0, a R. If we take inf a R : µ(e a ), then f a.e. 2
Proposition 7. If T is ergodic, then for f L p, p <, then A f : n T n f Lp Proof. Fro the ean ergodic theore, we know that A f Lp f with T f f. By the preceding propostion, f is constant a.e. Since A is bounded L p L p, it follows fro the doinated convergence theore and the fact that T is an isoetry L p L p, f f dµ A fdµ A fdµ fdµ n T n fdµ n fdµ fdµ.2 Maxial Ergodic Theore Lea 8. Suppose (, A, µ) (Z, P(Z), #), and T is the right-shift operator. Then if f L, sup T n f C f L, n L, Proof. Replacing f by f, we ay assue without loss of generality that f 0. We first observe that sup A f is easurable, since each A is easurable, being the linear cobination of easurable functions. We extend f on Z to f on R by f(x) : f(n) [n,n+), n Z For E Z, we define Ẽ R by Ẽ : n E [n, n+). ote that µ(ẽ) #(E), where µ denotes the Lebesgue easure on R, and f(x)dx R n Z f f L (Z). Moreover, L (R) Since + Since + 2, sup f(n + t)dt + 0 f(n + k) k k f(n + k) sup f(n + k) k + f(x + t)dt, x [n, n + ), ( + ( + f(n + t)dt ) + f(x + t)dt, x [n, n + ) + 0 ) + f(x + t)dt 2(Mf)(x), x [n, n + ), + 0 where Mf is the Hardy-Littlewood axial function on R. Applying the (, ) weak-type estiate for M, we see that ( # Z : sup T n f > µ x R : (Mf)(x) > ) 6 f L (R) 6 f L (Z) 2 n Proposition 9. (Maxial Ergodic Theore) Let (, A, µ, T ) be a easure-preserving syste. Then, for f L, sup T n f C f L n L, 3
Proof. Without loss of generality, assue that f 0. Furtherore, by the onotone convergence theore, it is enough to show that sup T n f C f L, 0 where C is a constant independent of 0. We define a function F : Z R by Then Since T n (T (x)) T n+ (x), we have F (x, n) A (f)(x) n L, (T n f)(x) n 0 n < (T n f)(x) n A (f)(t (x)) F (x, n) k F (x, n + ) Fix a, a 0, and set b a + 0. We define the function F b : Z R by F (x, n) n < b F b (x, n) 0 otherwise Then which iplies that A (f)(t (x)) k F b (x, n + ), 0, < a, n sup A (f)(t (x)) sup 0 F b (x, n + ) M(F b )(x, ), Set E x : sup 0 A (f)(x) >. Since T is easure-preserving, for any, µ( x : sup A (f)(t x) > ) µ(e ) 0 Hence, n n < a (µ #)( (x, ) Z : sup A (f)(t (x)) >, 0 < a ) aµ(e ) 0 By Fubini s theore and the estiate sup 0 A (f)(t (x)) M(F b )(x, ), n < a, (µ #)( (x, ) Z : sup A (f)(t (x)) >, 0 < a ) # Z : M(F b )(x, ) > dµ 0 6 F b (x, ) L (Z) dµ 6 b f(t n (x))dµ, where we use the weak-type estiate obtained in the preceding lea. Since f(t n (x))dµ f(x)dµ, we conclude that n aµ(e ) 6(a + 0) f L () ( ) a + 0 6 f L µ(e ) () a Letting a shows that µ(e ) 6 f L (). 4
.3 Pointwise Ergodic Theore Theore 0. Suppose f L (). Then for alost every x, the averages A (f) n f(t n (x)) converge to a it. Proof. Since µ(), L 2 (, A, µ) L (, A, µ) with f L 2 f L. Furtherore, by consider the n th truncates f n : f f n, we see that L 2 is dense in L. Let f L and ɛ > 0 be given. By the preceding rearks, we can write f f + f 2, where f L 2 and f 2 L < ɛ 2. Since the operator f T f is unitary, the proof of von euann s ergodic theore shows that the subspaces H T f L 2 : T f f, H diff T g g : g L 2 are orthogonal copleents. Hence, we can write f f,t + (T g g ) + h, where h L 2 < ɛ 2. Hence, there is a function f F + (T G G) L 2, where T F F such that f (F + (T G G)) L 2 < ɛ. Observe that A ( f) A (F + (T G G)) F + (T + G T G) I clai that (T + G G) 0 a.e. Indeed, since T is easure-preserving, n n 2 (G(T n (x)) 2 )dµ(x) G 2 L 2 (G(T n (x))) 2 so by the onotone convergence theore, n n 2 0, n by the continuity of x x. We conclude that A ( f)(x) F (x) n n 2 <, converges a.e., which iplies that G(T n (x)) n We now prove the convergence stateent for A (f). By the copleteness of R, it suffices to show that (A (f)(x)) is a Cauchy sequence for alost every x. Set E : x : sup A (f)(x) A M (f)(x) > 0,M 0 Since f f + h, where f L 2 and therefore A ( f)(x) 0 a.e., we have that ( ) µ x : sup A (h)(x) A M (h)(x) > \ x : sup A (f)(x) A M (f)(x) > 0, 0,M 0 0,M 0 hence it suffices to estiate the easure of E : x : 0 sup,m 0 A (h)(x) A M (h)(x) >. ote that E x : 2 sup A (h)(x) >, so by Proposition 9, µ (E ) µ(e ) µ a.e. ( ) x : 2 sup A (h)(x) > 2 6 h L 2 h L 2 2ɛ Since ɛ > 0 was arbitrary, we conclude that µ(e ) 0, which shows that (A (f)(x)) is Cauchy for alost every x. Corollary. (Pointwise Ergodic Theore) If f L (, A, µ) and T is ergodic, then A (f)(x) fdµ a.e. Proof. Since T is ergodic, Proposition 4 shows that A f f L 0,, where f fdµ. It is a consequence of the Borel-Cantelli Lea that we can choose a subsequence (A k f) such that A k (f)(x) f (x) for alost every x. By the preceding proposition and the uniqueness of its, we conclude that A (f)(x) fdµ a.e. 5
.4 Application to Law of Large ubers Lea 2. Let (Ω i, A i, µ i ) be a collection of identical probability spaces. Set Ω Ω i, let A be the cylinder σ-algebra, and let µ be the product easure. Let T : Ω Ω be the shift Then T is an ergodic transforation. T ((x, x 2, )) (x 2, x 3, ) Proof. We first verify that T is easure-preserving. Since the collection of sets E for which µ(t (E)) µ(e) fors a σ-algebra, it suffices to show that µ(t (E)) µ(e), where E A is a cylinder set. Fix a cylinder set E, and let the axial index such that π (E) Ω. Then E E i i+ Ω, where E i A i. It is evident that ( ) T (E) Ω E i Ω, µ(t (E)) µ E i µ(e) i+ By the sae arguent, it suffices to show that T (E) E µ(e) 0, for a cylinder set E A. Suppose T (E) E and µ(e) 0. Then E i i+ Ω E T (E) Ω E i i+ which iplies that E Ω, E i+ E i, i E Ω, which has easure. Proposition 3. Let ( n ) n be a sequence of i.i.d. rando variables on a probability space (Ω, F, P) such that E[ ] <. Then Ω, i E[ ], alost surely Proof. Consider the probability space (R, B(R ), µ), where µ is the product easure on B(R ) induced by the laws of the i. Define f : R R by f(x, x 2, ) f(x ) Let T : R R be the shift operator defined above. So (R, B(R ), µ, T ) is a easure-preserving syste. I clai that f L. Indeed, By the pointwise ergodic theore, x i By definition of the easure µ, it follows f(t ((x i ) )) fdµ E[ ] R alost surely i f(t (( i ) )) E[ ] alost surely References [] Stein, Elias, and Rai Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces. Princeton University Press, 2005. 6