Page 126 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 10: Measurable Functions

Haracteristic Function: Let A be subset of real numbers. We define the characteristic function Notes
A
of the set A as follows:
1 if x A
(x) =
A 0 if x A

Lebesgue Measurable Function: A function f : E R* is said to be a Lebesgue measurable function
on E or a measurable function on E iff the set
–1
E (f > ) = {x E : f (x) > } = f { , )} is a measurable subset of E R.
Measurable Set: A set E is said to be measurable if for each set T, we have
c
m* (T) = m* (T E) + m* {T E )
+
Non-negative Functions: Let f be a function, then its positive part, written f and its negative
–1
part, written f , are defined to be the non-negative functions given by
+
–1
f = max (f, 0) and f = max (–f, 0) respectively.
Riesz Theorem: Let {f } be a sequence of measurable functions which converges in measure to f.
n
Then there is a subsequence f which converges to f a.e.
n k
Simple Function: A real valued function is called simple if it is measurable and assumes only
a finite number of values.
If is simple and has the values , , … , then
1 2 n
n
= i A i
i 1
where A = {x : (x) = }
i i
and A A is a null set.
i j
Step Function: A real valued function S defined on an interval [a, b] is said to be a step function
if these is a partition a = x < x … < x = b such that the function assumes one and only one value
o 1 n
in each interval.
Subsequence: If (x ) is a given sequence in X and (n ) is an strictly increasing sequence of positive
n k
integers, then x is called a subsequence of (x ).
n k n

10.4 Review Questions

1. If f is a measurable function and c is a real number, then is it true to say that cf is measurable?

2. A non-zero constant function is measurable if and only if X is measurable comment.
3. Let Q be the set of rational number and let f be an extended real-valued function such that
{x : f (x) > } is measurable for each Q. Then show that f is measurable.

4. Show that if f is measurable then the set {x : f (x) = } is measurable for each extended real
number .
5. If f is a continuous function and g is a measurable function, then prove that the composite
function fog is measurable.

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