28-1
NOTE
,
*) is a measurable space.
Proposition
Suppose are all measurable functions. Then
i)
ii)
iii)
iv)
v) ,
NOTE
i), ii) vector space
ii) if
i)
Choose such that
iii)
NOTE
is a decreasing sequence.
Theorem
Suppose is a sequence of measurable functions, . Then following are measurable:
i)
ii)
iii) If pointwise, then is also measurable.
Proof
Definition (Simple Function)
is called a simple function if there exists a finite disjoint collection of sets in and some real numbers ,
NOTE
i) Suppose , . Then
ii) is measurable
Proposition
is simple if and only if is measurable and the range of is finite.
Set
is measurable in .
NOTE
Step functions are simple functions.
Definition
If is a simple function with representation , then we define the integral of with respect to as
Let be a measure space
NOTE
If not , then let , and , then will happen so .