Page 341 - DMTH401_REAL ANALYSIS

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

Notes
Figure 28.3

Here, f looks like a “V” and is bounded above by 1. The top figures show partitions of the range
of f into halves, quarters, then eights and the bottom figures show the corresponding simple
functions. It is clear that s  s  s .
1 2 3
Using Theorem 2 on limits of simple functions, it is easy to prove that measurable functions are
closed under all the usual arithmetic operations. Of course, the proofs aren’t particularly difficult
to prove directly.
p
Theorem 3: If f and g are measurable, then f + g, f  g, 1/f, and |f| where p > 0, are also measurable,
whenever each expression is defined.
Proof: We need to add the last statement for f + g and 1/f. For 1/f we need f to never vanish and
for f + g we don’t want f(x) + g(x) to give a non-sense statement such as ¥ – ¥ or –¥ + ¥ at any
point x  X.
p
The proofs that f + g, f g, 1/f, and |f| are measurable are all the same: we just show that each
combination can be written as a limit of simple functions. By Theorem 2 we can write f = lim s
n
and g = lim t for simple functions s , t , n = 1, 2, 3, ... . Therefore,
n n n
f + g = lim(s + t )
n n
and
f g = lim(s t ).
n n
Since the sum and product of simple functions are simple, it follows that f + g and f g are limits
of simple functions, so are measurable.

p
To see that 1/f and |f| are measurable, write the simple function s as a finite sum
n
s = å a c ,
n nk A nk
k
where A , A ,...  S are finite in number and pairwise disjoint, and a , a ,...  , which we may
n1 n2 n1 n2
assume are all non-zero. If we define
1
-
u = å a c and v = å a p c ,
n nk A nk n nk A nk
k k
which are simple functions, then a short exercise shows that
–1
p
f = lim u and |f| = lim v ,
n n
–1
p
where in the first equality we assume that f is nonvanishing. This shows that f and |f| are
measurable.
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