Page 345 - DMTH401_REAL ANALYSIS

Unit 28: Sequences of Functions and Littlewood’s Third Principle

(iii) Now find a closed set C  K such that m(K \C) < . Show that m(X\C) < 3 and the Notes
2 2
restriction of f to C is a continuous function.
(iv) Finally, prove Luzin’s theorem dropping the assumption that f is non-negative.

2. A sequence {f } of real-valued measurable functions is said to be convergent in measure if

n
there is a measurable function f such that for each  > 0,
-
lim  ({x; f (x) f(x) ³  }) = 0.
n
n®¥
(Does this remind you of the weak law of large numbers?) Prove that if {f } converges in
n
measure to a measurable function f, then f is a.e. real-valued, which means {x; f(x) = ±¥} is
measurable with measure zero. If {f } converges to two functions f and g in measure, prove
n
that f = g a.e. Suggestion: To see that f = g a.e., prove and then use the “set-theoretic triangle
inequality”: For any real-valued measurable functions f, g, h, we have
{  } {  }
{x; |f(x) – g(x)| ³}  x; f(x) – h(x) ³ 2  x; h(x) – g(x) ³ 2 .

3. Here are some relationships between convergence a.e., a.u., and in measure.
(a) (a.u.  in measure) Prove that if f ® f a.u., then f ® f in measure.
n n
(b) (a.e.  in measure) From Egorov’s theorem prove that if X has finite measure, then
any sequence {f } of real-valued measurable functions that converges a.e. to a real-
n
valued measurable function f also converges to f in measure.
(c) (In measure  a.u. nor a.e.) Let X = [0,1] with Lebesgue measure. Given n  , write
k
k
n = 2 + i where k = 0, 1, 2,... and 0  i < 2 , and let f be the characteristic function of
n
+
é i i 1ù
the interval ê , k ú . Draw pictures of f , f , f ,...,f . Show that f ® 0 in measure,
ë 2 k 2 û 1 2 3 7 n
but lim f (x) does not exist for any x  [0, 1]. Conclude that {f } does not converge to
n®¥ n n
f a.u. nor a.e.
4. A sequence {f } of real-valued, measurable functions is said to be Cauchy in measure if for
n
any  > 0,
 ({x; f (x) – f (x) ³  ) } ® 0, as n, m ® ¥.
m
n
Prove that if f ® f in measure, then {f } is Cauchy in measure.
n n
5. In this problem we prove that if a sequence {f } of real-valued measurable functions is
n
Cauchy in measure, then there is a subsequence {f } and a real-valued measurable function
n k
f such that f ® f a.u. Proceed as follows.
n k
(a) Show that there is an increasing sequence n < n <  such that
1 2
1
 ({x; f (x) – f (x) ³  ) }) < , for all n, m ³ n .
k
m
n
2 k
(b) Let
m { 1 }
¥
A =  x; f (x) – f n (x) ³ k .
n
=
k m k k +1 2
c
Show that {f } is a Cauchy sequence of bounded functions on the set A . Deduce that
n k m
c
there is a real-valued measurable function f on A :=  ¥ m 1 A such that {f } converges
=
m
n k
c
uniformly to f on each A .
m
c
(c) Define f to be zero on A . Show that f ® f a.u.
n
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