00000191.htm

172 XIII INTEGRATION

Show that the series (p^iff + <pA₂# + ---- 1- <p*_n9 -h • • • converges almost everywhere.
(Observe that

*f- ± U*

fc=0

almost everywhere; integrate both sides, and use (13.8.5).)

(b) Let B be the measurable set of points x e X such that (Uⁿ - g)(x) > 0 for at

least one integer n ^ 0. Show that in B the sequence of functions

converges to 0 almost everywhere. (Deduce from (a) above that the intersection of
Q A_n with the set (x e X : g(x) > 0} is negligible. Now replace g and /successively by

U^m -g and U^m •/, where m ^ 1.)

let

+ (U < gm + • - • + (t/-¹ • g)(x)
and let R*(/, g) = sup | R_n(/, g) \ . Show that R*(/, g)(x) < + oo for almost all x e B_a .

(For each t > 0, let A_t be the set of points x e B₀ such that R*(/, g)(x) > t. Show that
*f 9 dp ^ Ni(/) by applying Problem 16(d) to |/| - t • g.)

(d) Let $ e «S?i be such that ®(x) > 0 for all x e X (cf. (1 3.1 5.7)). Show that the set
Xoo of points x e X such that

®(x) + (£/• <£)(*) + ••• + (£/"• <&)(*) + ••• = + oo

does not depend on the choice of <1>, but only on U.
For each x e X, let

G_n(x) = $(*) + ( U • <S>)(x) +•
and let G(x) = lim G_n(x) (so that GW = -f oo for x £ Xoo). Show that for each

n-*cG

integrable set A c: Xoo and each f > 0 we have t - (U • <p_A) ^ G, and hence, that the
function U - <p_A is zero almost everywhere in X₀ = CX°Q . (Observe that, if A_n is the
set of x e A such that G_n(;c) > t, then *9?_An <; G_n .)

12. THE SPACE L^w

If/is a mapping of X into R, the maximum in measure (resp. minimum in
measure) of/on X (with respect to the measure p) is defined to be the greatest
lower bound (resp. least upper bound) of the real numbers a such that
f(x) ^ a (resp. f(x) ^ d) almost everywhere with respect to \JL, and is denoted
by M*>(/), or ess sup/, or ess sup/(x) (resp. mJJ), or ess inf/, or ess inf/(;c)).

Clearly we have m₀₀(/) = -MJ-/).



	172 XIII INTEGRATION Show that the series (p^iff + <pA₂# + ---- 1- <p_n9* -h • • • converges almost everywhere. (Observe that f- ± U fc=0 almost everywhere; integrate both sides, and use (13.8.5).) (b) Let B be the measurable set of points x e X such that (Uⁿ - g)(x) > 0 for at least one integer n ^ 0. Show that in B the sequence of functions

	converges to 0 almost everywhere. (Deduce from (a) above that the intersection of Q A_n with the set (x e X : g(x) > 0} is negligible. Now replace g and /successively by U^m -g and U^m •/, where m ^ 1.) (c) Let B₀ be the measurable set of points x e X such that g(x) > 0. For each x e B₀ „ let

	+ (U < gm + • - • + (t/-¹ • g)(x) and let R(/, g) = sup \| R_n(/, g) \* . Show that R(/, g)(x) <* + oo for almost all x e B_a . n (For each t > 0, let A_t be the set of points x e B₀ such that R(/, g)(x) > t.* Show that f 9 dp* ^ Ni(/) by applying Problem 16(d) to \|/\| - t • g.) (d) Let $ e «S?i be such that ®(x) > 0 for all x e X (cf. (1 3.1 5.7)). Show that the set Xoo of points x e X such that ®(x) + (£/• <£)() + ••• + (£/"• <&)() + ••• = + oo

	does not depend on the choice of <1>, but only on U. For each x e X, let

	G_n(x) = $() + ( U • <S>)(x)* +• and let G(x) = lim G_n(x) (so that GW = -f oo for x £ Xoo). Show that for each n-cG integrable set A c: Xoo and each f > 0 we have t - (U •* <p_A) ^ G, and hence, that the function U - <p_A is zero almost everywhere in X₀ = CX°Q . (Observe that, if A_n is the set of x e A such that G_n(;c) > t, then *9?_An <; G_n .)

	12. THE SPACE L^w If/is a mapping of X into R, the maximum in measure (resp. minimum in measure) of/on X (with respect to the measure p) is defined to be the greatest lower bound (resp. least upper bound) of the real numbers a such that f(x) ^ a (resp. f(x) ^ d) almost everywhere with respect to \JL, and is denoted by M>(/), or ess sup/, or ess sup/(x) (resp. mJJ),* or ess inf/, or ess inf/(;c)). Clearly we have m₀₀(/) = -MJ-/).