Measure, Integration & Real Analysis, 2021a

80 Chapter 3 Integration
Now we can show that our definition
of integration does the right thing with
simple measurable functions that might
not be expressed in the standard representation.
The result below differs from 3.7
mainly because the sets E 1 ,...,E n in the
result below are not required to be disjoint.
Like the previous result, the next
result would follow immediately from the
linearity of integration if that property had
already been proved.
If we had already proved that
integration is linear, then we could
quickly get the conclusion of the
previous result by integrating both
sides of the equation
∑ m j=1 a jχ
Aj
= ∑ n k=1 b kχ
Bk
with
respect to μ. However, we need the
previous result to prove the next
result, which is used in our proof
that integration is linear.
3.15 integral of a linear combination of characteristic functions
Suppose (X, S, μ) is a measure space, E 1 ,...,E n ∈S, and c 1 ,...,c n ∈ [0, ∞].
Then
∫ ( n ) n
∑ c k χ
Ek
dμ = ∑ c k μ(E k ).
k=1 k=1
Proof The desired result follows from writing the simple function ∑ n k=1 c kχ
Ek
in
the standard representation for a simple function and then using 3.7 and 3.13.
Now we can prove that integration is additive on nonnegative functions.
3.16 additivity of integration
Suppose (X, S, μ) is a measure space and f , g : X → [0, ∞] are S-measurable
functions. Then ∫
∫ ∫
( f + g) dμ = f dμ + g dμ.
Proof The desired result holds for simple nonnegative S-measurable functions (by
3.15). Thus we approximate by such functions.
Specifically, let f 1 , f 2 ,... and g 1 , g 2 ,... be increasing sequences of simple nonnegative
S-measurable functions such that
lim f k(x) = f (x) and lim g k (x) =g(x)
k→∞ k→∞
for all x ∈ X (see 2.89 for the existence of such increasing sequences). Then
∫
∫
( f + g) dμ = lim ( f k + g k ) dμ
k→∞
= lim f k dμ + lim
∫ ∫
= f dμ + g dμ,
k→∞
∫
k→∞
∫
g k dμ
where the first and third equalities follow from the Monotone Convergence Theorem
and the second equality holds by 3.15.
Measure, Integration & Real Analysis, by Sheldon Axler