Page 119 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Theorem 4: If f is a measurable function and f = g a.e. then g is measurable.
Proof: Let E = {x : f (x) g (x)}.
Then m (E) = 0
Let be a real number.

{x : g (x) > } = {x : f (x) > } {x E : g (x) > } – {x E : g (x) }
since f is measurable, the first set on the right is measurable i.e. {x : f (x) > } is measurable.
The last two sets on the right are measurable since they are subsets of E and m (E) = 0.
Thus, {x : g (x) > } is measurable.

So, g is measurable.

Example: Give an example of function for which f is not measurable but |f| is measurable.
Sol: Let k be a non-measurable subset of E = [0, 1).
Define a function f : E R by

1 if x k
f (x) =
1 if x k
The function f is not measurable, since E (f > 0) (=k) is a non-measurable set. But |f| is measurable
as the set
E if 1
E (|f| > ) = is measurable
if 1

10.1.5 Characteristic Function

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

Note The characteristic function of the set A is also called the indicator function of A.
A
Theorem 5: Show that the characteristic function is measurable iff A is measurable.
A
Proof: Let be measurable.
A
Since A = {x : (x) > 0} is measurable.
A
But is measurable, therefore the set {x : (x) > 0} is measurable.
A A
A is measurable.
Conversely, let A be measurable and be any real number.
if 1
then E ( > ) = A if 0 1
A
E if 0

112 LOVELY PROFESSIONAL UNIVERSITY