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9 MEASURABLE FUNCTIONS 147

12. (a) Let (//)o 5=^/1 be a sequence of real numbers, and let m be an integer fS/z. Let
L_m be the set of indices / with the following property : there exists an integer p such
that 0 fg;? <; w and t_t + t_l+1 -\ ----- h fi+_P-i ^ 0. Show that Y t_t *> 0. (Observe that

t€L_m

if i e L_m , and if /? is the smallest of the integers with the above property, then
i + 1, . . . , i -f p ~ 1 belong to L_m .)

(b) Let X be a locally compact space and p, a positive measure on X. Let u : X -> X
be a proper continuous mapping such that U(JJL) = p.. If /is any ^-integrable function
on X, put/o =/and/t =/o u^k (k ]> 1). For each integer w, let A_m be the measurable
set of points x e X such that one of the sums/₀(jt) + /i(x) -f • • • + /_P(x), where/? ^ w,

is ^> 0. Show that /(x) <tf/x(x) ^ 0. (For each integer n > 0 and each x e X, consider

jA_m

the sequence (/((jc))o^^_n+_m , and apply (a) to this sequence, denoting the correspond-
ing set of indices by L_m(x). For each k <! n + m, let B_fc be the set of x e X such that

A: e L_m(x), and deduce from (a) that Y /_A(x) du(x) ^> 0. Now use the fact that

* = ₀J_BkB_ft = u~^k(A_m) for 0 g A: g « to deduce that

(« + 1) f /W ^W + m f |/(*)| ^W ^ 0

JA»n J

for all n.) If A is the union of the A_m , conclude that f(x) dp,(x) ^ 0 (" maximal

ergodic theorem").

1 "-¹a < lim sup - X /*(*)

n-*oo M fc = 0

for all x e C. Show that aju-(C) <; f |/(x)| dp(x). (Apply (b) to the function /- a<p_c .)

(d) Let a, b be two real numbers such that a < b. If E is the set of points x e X
such that

I n-l lⁿ~*

lim inf - ]£ /_fc(x) < a < 6 < lim sup - V /*(*),

n-foo n * = 0 n-+oo W fc« 0

show that E is ^-negligible. (Deduce first from (c) that E is integrable, and then apply
(b) to each of the functions (/— b)<p_E and (a -~~f)(pE, using the fact that w(E) <= E.)
Hence show that, for almost all XE X, the sequence

/I "-¹ \

(-Z/(«*(*»)

\n*.o /

converges to a limit /*(x), that /* is integrable and that /*(w(x))=/*(x) almost
everywhere (G. D. Birkhoff's ergodic theorem). (Take a, b to be all pairs of rational
numbers such that a<b.)

(e) If X is compact, show that I /* d^ = fd^. (Reduce to the case /2> 0, and
consider first the case where/is bounded; then pass to the general case by observing



	9 MEASURABLE FUNCTIONS 147 12. (a) Let (//)o 5=^/1 be a sequence of real numbers, and let m be an integer fS/z. Let L_m be the set of indices / with the following property : there exists an integer p such that 0 fg;? <; w and t_t + t_l+1 -\ ----- h fi+_P-i ^ 0. Show that Y t_t >* 0. (Observe that *t€L_m* if i e L_m , and if /? is the smallest of the integers with the above property, then i + 1, . . . , i -f p ~ 1 belong to L_m .) (b) Let X be a locally compact space and p, a positive measure on X. Let u : X -> X be a proper continuous mapping such that U(JJL) = p.. If /is any ^-integrable function on X, put/o =/and/t =/o u^k (k ]> 1). For each integer w, let A_m be the measurable set of points x e X such that one of the sums/₀(jt) + /i(x) -f • • • + /_P(x), where/? ^ w, is ^> 0. Show that /(x) <tf/x(x) ^ 0. (For each integer n > 0 and each x e X, consider jA_m the sequence (/((jc))o^^_n+_m , and apply (a) to this sequence, denoting the correspond- ing set of indices by L_m(x). For each k <! n + m, let B_fc be the set of x e X such that

	A: e L_m(x), and deduce from (a) that Y /_A(x) du(x) ^> 0. Now use the fact that * = ₀J_BkB_ft = u~^k(A_m) for 0 g A: g « to deduce that (« + 1) f /W ^W + m f \|/()\| ^W ^ 0 JA»n J for all n.)* If A is the union of the A_m , conclude that f(x) dp,(x) ^ 0 (" maximal ergodic theorem"). (c) Let a be a real number and C an integrable set such that 1 "-¹a < lim sup - X /() n-oo M fc = 0* for all x e C. Show that aju-(C) <; f \|/(x)\| dp(x). (Apply (b) to the function /- a<p_c .) (d) Let a, b be two real numbers such that a < b. If E is the set of points x e X such that *I n-l* lⁿ~* lim inf - ]£ /_fc(x) < a < 6 < lim sup - V /(), *n-foo n = 0** n-+oo W fc« 0 show that E is ^-negligible. (Deduce first from (c) that E is integrable, and then apply (b) to each of the functions (/— b)<p_E and (a -~~f)(pE, using the fact that w(E) <= E.) Hence show that, for almost all XE X, the sequence

	/I "-¹ \ (-Z/(«(») \n*.o /

	converges to a limit /(x), that / is integrable and that /(w(x))=/(x) almost everywhere (G. D. Birkhoff's ergodic theorem). (Take a, b to be all pairs of rational numbers such that a<b.) (e) If X is compact, show that I /* d^ = fd^. (Reduce to the case /2> 0, and consider first the case where/is bounded; then pass to the general case by observing