Page 338 - DMTH401_REAL ANALYSIS

Real Analysis

Notes and in general,
 = sup a = inf{a , a , a ,...}.
n k n n+1 n+2
³
k n
Note that
     ···     ···
1 2 3 n n+1
is an non-decreasing sequence since each successive  is obtained by taking the infimum of a
n
smaller set of elements. Since { } is an non-decreasing sequence, the limit lim  exists, and
n n
equals supn  . We define the lim inf of the sequence {a } as
n n
lim inf a := sup  = lim  = lim (inf{a , a , a ,...}).
n n n n®¥ n n+1 n+2
n
Note that the term “lim inf” of {a } fits well because lim inf a is the limit of a sequence of
n n
infimums.
Example: For the sequence a shown in Figure 28.1, we have
n
 = a ,  = a ,  = a ,  = a ,  = a ,...,
1 2 2 2 3 4 4 4 5 6
so lim inf a is exactly the limit of the even-indexed a ’s.
n n
The following lemma contains some useful properties of limsup’s and liminf’s. Since its proof
really belongs in a lower-level analysis.
Lemma: Let A   be non-empty and let {a } be a sequence of extended real numbers.
n
1. sup A = –inf(–A) and inf A = –sup(–A), where –A = {–a; a  A}.
2. lim sup a = –lim inf(–a ) and lim inf a = –lim sup(–a ).
n n n n
3. lim a exists as an extended real number if and only if lim sup a = lim inf a , in which case,
n n n
lim a = lim sup a = lim inf a .
n n n
4. If {b } is another sequence of extended real numbers and a  b for all n sufficiently large,
n n n
then
lim inf a  lim inf b and lim sup a  lim sup b .
n n n n
28.2 Operations on Measurable Functions

Let {f } be a sequence of extended-real valued functions 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 pointwise
n n n n
to the sequence of extended real numbers {f (x)} at each point x  X. For example,
n
lim sup f : X ® 
n
is the function defined by
(lim sup f )(x) := lim sup(f (x)) at each x  X.
n n
We define the limit function lim f by
n
(lim f )(x) := lim (f (x))
n n®¥ n
at those points x  X where the right-hand limit exists.
We now show that limiting operations don’t change measurability.

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