Page 131 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes This representation of is called the canonical representation. Here E ’s are disjoint and ’s are
i i
finite in number, distinct and non-zero.
Elementary Integral

Definition: If vanishes outside a set of finite measure, we define the elementary integral of by
n
(x) dx i mE when has the canonical representation.
i
i 1
n
= i E i .
i 1

We sometimes abbreviate the expression for this integral . If E is any measurable set, we

define the elementary integral of over E by .
E
E
b
If E = [a, b], then the integral will be denoted by .
[a, b] a
Theorem 1: Let and be simple functions which vanish outside a set of finite measure, then

(a b ) = a b and if a.e., then

Proof: Since and are simple functions.
Therefore these can be written in the canonical form

m
= i A i

m
and = B j B j
j 1
where {A } and {B} are disjoint sequences of measurable sets and
i j
A = {x : (x) = }
i i
and B = {x : (x) = }
j j
The set E obtained by taking all intersections A B form a finite disjoint collection of measurable
k i j
sets. We may write
N
= a k E k and
k 1

N
= b k E k (where N = mm )
k 1

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