Page 327 - DMTH401_REAL ANALYSIS

Unit 27: Measurable Functions and Littlewood’s Second Principle

Notes
Example: Let X = S , where S = {0,1}, be the sample space for a Monkey-Shakespeare

experiment (or any other experiment involving a sequence of Bernoulli trials). Let f : X  [0, ]
be the number of times the Monkey types sonnet 18:
f(x , x , x ,...) = the number of i’s such that x = 1.
1 2 3 i
Notice that f =  when the Monkey types sonnet 18 an infinite number of times (in fact, as we see
that f =  on a set of measure). To show that f is measurable, write f as

f = lim f ,
n
n
where f is the number of i’s in 1, 2,..., n such that x = 1. Notice that f £ f £ f £  are non-
i i 1 2 3
decreasing, so it follows that for any a  ,

f(x) £ a  f (x) £ a for all n  x   {f £ a}. {f < a}.
n n n
=
n 1
Thus,

1
f [–, a] =  f [- , a].
-
–1
n
n 1
=
n
n
The set {f £ a} is of the form A  S  S  S   where A  S is the subset of S consisting of those
n n n
points with no more than a total of a entries with 1’s. In particular, {f £ a}  R (C ) and hence, it
n
belongs to S (C ). Therefore, {f £ a} also belongs to S (C ), so f is measurable.
We shall return to this example when we study limits of measurable functions.
As we defined simple functions. For a quick review in the current context of our -algebra, S ,
recall that a simple function (or S -simple function to emphasize the -algebra, S ) is any function
of the form
N
s = å a  A ,
n
n 1 n
=
where a ,..., a   and A ,..., A S are pairwise disjoint. We know that we don’t have to take
1 N 1 N
the A ’s to be pairwise disjoint, but for proofs it’s often advantageous to do so.
n
Theorem 1: Any Simple Function is Measurable
Proof: Let s = å N n 1 a  A be a simple function where a ,..., a   and A ,..., A  S are pairwise
=
n
1
N
1
n
N
disjoint. If we put A = X \ {A    A ) and a = 0, then
N+1 1 N N+1
X = A  A    A  A ,
1 2 N N+1
a union of pairwise disjoint sets, and s = a on A , for each n = 1, 2,..., N + 1. It follows that
n n
N 1
+
s [a, ] = {x  X; s(x) > a} =  {x A ; s(x) a}
>

–1
n
=
n 1
+
N 1
=  {x A ; a > a}.

n
n
=
n 1
Since
ì A n if a > a
n
{x  A ; a > a} = í
n n Æ otherwise.
î
–1
it follows that s [a, ] is just a union of elements of S . Thus, s is measurable.
LOVELY PROFESSIONAL UNIVERSITY 321