Page 339 - DMTH401_REAL ANALYSIS

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

Theorem 1: Limits preserve measurability Notes
If {f } is a sequence of measurable functions, then the functions
n
sup f , inf f , lim sup f , and lim inf f
n n n n
are all measurable. If the limit lim f (x) exists at each x  X, then the limit function lim f is
n®¥ n n
measurable. For instance, if the sequence {f } is monotone, that is, either non-decreasing or
n
non-increasing, then lim f is everywhere defined and it is measurable.
n
–1
Proof: To prove that sup f is measurable, we just have to show that (sup f ) [–¥, a]  S for each a
n n
 . However, this is easy because by definition of supremum, for any a  ,
sup{f (x), f (x), f (x),...}  a  f (x)  a for all n,
1 2 3 n
therefore
¥
(sup f ) [–¥, a] = {x; sup f (x)  a} =  {x; f (x) a}
–1

n n n
=
n 1
¥
-
1
=  f [-¥ ,a].
n
=
n 1
1
–1
-
Since each f is measurable, we have f [-¥ ,a]  S , so (sup f ) [–¥, a]  S as well. Using an
n n n
analogous argument one can show that inf f is measurable.
n
To prove that lim sup f is measurable, note that by definition of lim sup,
n
lim sup f : = inf s ,
n n n
where s = sup f . Since the sup and inf of a sequence of measurable functions are measurable,
n k³n k
we know that s is measurable for each n and hence lim sup f = inf s is measurable. An analogous
n n n n
argument can be used to show that lim inf f is measurable (just note that lim inf f = sup  where
n n n
 = inf f ).
n k³n k
If the limit function lim f is well-defined, then by Part (3) of above Lemma we know that lim f =
n n
lim sup f (= lim inf f ). Thus, lim f is measurable.
n n n
In particular, if f is a function on X and if f = lim s , where the s ’s are simple function (which are
n n
measurable by Theorem 3.4), then f is measurable.
¥
Example: Let X = S , where S = {0,1}, a sample space for the Monkey-Shakespeare
experiment (or any other sequence of Bernoulli trials), and let f : X ® [0, ¥] be the random
variable given by the number of times the Monkey types sonnet 18. Then
¥
f(x) = å x n
n 1
=
That is,
¥ n
f = å c A = lim å c A ,
n 1 n n®¥ k 1 k
=
=
where A = S  S    S  {1}  S  S   where {1} is in the n-th slot. Since each A is measurable,
n n
it follows that each c is measurable and hence so is f.
A n
Given f: X ® , we define its non-negative part f : X ® [0, ¥] and its non-positive part f : X ®
+ –
[0, ¥] by
f := max{f, 0} = sup{f, 0} and f := – min{f, 0} = – inf{f, 0}.
+ –
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