Page 343 - DMTH401_REAL ANALYSIS

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

lim (B ) = 0. Notes
m®¥ m
Choose N such that (B ) <  and let B = B . Then by definition of B , one can check that holds.
N N N
This concludes Step 1.
Step 2: We now finish the proof. Let  > 0. Then by Step 1, for each k   we can find a measurable
set A  X and a corresponding natural number N   such that
k k
 1
(A ) < and for x A c , |f(x) – f (x)| < for all n > N .
k k k n k
2 k
Now put A =  ¥ k 1 A . Then (A) <  and we claim that f ® f uniformly on A . Indeed, let  > 0 and
c
k
=
n
choose k   such that 1/k < . Then
¥
x  A =  A  x A c k
c

c
j
=
j 1
1
 |f(x) – f (x)| < for all n > N
n k
k
 |f(x) – f (x)| <  for all n > N .
n k
Thus, f ® f a.u.
n
We remark that one cannot drop the finiteness assumption.
Self Assessment
Fill in the blanks:
1. Let {f } be a sequence of ........................................... on a measure space (X, S , ). We define
n
the functions sup f , inf f , lim sup f , and lim inf f , by applying these limit operations
n n n n
pointwise to the sequence of extended real numbers {f (x)} at each point x  X.
n
2. If the sequence {f } is ................................, that is, either non-decreasing or non-increasing,
n
then lim f is everywhere defined and it is measurable.
n
p
3. If f and g are measurable, then .............................., and |f | where p > 0, are also measurable,
whenever each expression is defined.
4. Every convergent series of measurable functions is ................................... when certain sets
of measure  are neglected, where  can be as small as desired.

28.4 Summary

 For a sequence {a } of extended real numbers, we know, in general, that lim a does not
n n
exist; for example, it can oscillate. Assuming that the sequence continues the way it looks
like it does, it is clear that although limit lim a does not exist, the sequence does have an
n
“upper” limiting value, given by the limit of the odd-indexed a ’s and a “lower” limiting
n
value, given by the limit of the even-indexed a ’s. Now how do we find the “upper” (also
n
called “supremum”) and “lower” (also called “infimum”) limits of {a }? It turns out there
n
is a very simple way to do so, as we now explain.
 Let {f } be a sequence of extended-real valued functions on a measure space (X, S , ). We
n
define the functions sup f , inf f , lim sup f , and lim inf f , by applying these limit operations
n n n n
pointwise to the sequence of extended real numbers {f (x)} at each point x  X. For example,
n

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