Real and Complex Analysis (Rudin)

74 REAL AND COMPLEX ANALYSIS
18 Let II. be a positive measure on X. A sequence {!.} of complex measurable functions on X is said
to converge in measure to the measurable functionf if to every € > 0 there corresponds an N such that
J1.({x: If.(x) - f(x)l > €}) < €
for all n > N. (This notion is of importance in probability theory.) Assume J1.(X) < 00 and prove the
following statements:
(a) Iff.(x) ..... f(x) a.e., then!. ..... f in measure.
(b) Iff. e I!(J1.) and Ilf. - flip ..... 0, then!. ..... f in measure; here 1 $ P $ 00.
(c) If!. ..... f in measure, then {I.} has a subsequence which converges to f a.e.
Investigate the converses of (a) and (b). What happens to (a), (b), and (c) if J1.(X) = 00, for
instance, if II. is Lebesgue measure on R I ?
19 Define the essential range of a function fe L""(J1.) to be the set RI consisting of all complex
numbers w such that
J1.({x: If(x) - wi < €}) > 0
for every € > o. Prove that R I is compact. What relation exists between the set R I and the number
IIfll"" ?
Let A I be the set of all averages
where E e 9Jl and J1.(E) > o. What relations exist between AI and RI? Is AI always closed? Are there
measures II. such that AI is convex for every fe L""(J1.)? Are there measures II. such that AI fails to be
convex for some f e L""(J1.)?
How are these results affected if L""(J1.) is replaced by ll(J1.), for instance?
20 Suppose cp is a real function on R I such that
cp(f f(x) dX) $ f cp(f) dx
for every real bounded measurable f Prove that cp is then convex.
21 Call a metric space Y a completion of a metric space X if X is dense in Y and Y is complete. In
Sec. 3.15 reference was made to "the" completion of a metric space. State and prove a uniqueness
theorem which justifies this terminology.
22 Suppose X is a metric space in which every Cauchy sequence has a convergent subsequence. Does
it follow that X is complete? (See the proof of Theorem 3.11.)
23 Suppose II. is a positive measure on X, J1.(X) < oo,Je L""(J1.), Ilfll"" > O,and
(n = 1, 2, 3, ... ).
Prove that
lim IX. + I = II !II"".
,.-co IX,.
24 Suppose II. is a positive measure,f e I!(J1.), g e I!(J1.).
(a) If 0 < p < 1, prove that
that I:l(f, g) = J If - g IP dJi. defines a metric on I!(J1.), and that the resulting metric space is complete.