Measure, Integration & Real Analysis, 2021a

Section 3A Integration with Respect to a Measure 83
Now we prove that integration with respect to a measure has the additive property
required for a good theory of integration.
3.21 additivity of integration
Suppose (X, S, μ) is a measure space and f , g : X → R are S-measurable
functions such that ∫ | f | dμ < ∞ and ∫ |g| dμ < ∞. Then
∫
∫
( f + g) dμ =
∫
f dμ +
g dμ.
Proof Clearly
( f + g) + − ( f + g) − = f + g
= f + − f − + g + − g − .
Thus
( f + g) + + f − + g − =(f + g) − + f + + g + .
Both sides of the equation above are sums of nonnegative functions. Thus integrating
both sides with respect to μ and using 3.16 gives
∫
∫ ∫ ∫
∫ ∫
( f + g) + dμ + f − dμ + g − dμ = ( f + g) − dμ + f + dμ + g + dμ.
Rearranging the equation above gives
∫
∫
∫
( f + g) + dμ − ( f + g) − dμ =
∫
f + dμ −
∫
f − dμ +
∫
g + dμ −
g − dμ,
where the left side is not of the form ∞ − ∞ because ( f + g) + ≤ f + + g + and
( f + g) − ≤ f − + g − . The equation above can be rewritten as
∫
∫ ∫
( f + g) dμ = f dμ + g dμ,
Gottfried Leibniz (1646–1716)
invented the symbol ∫ to denote
completing the proof.
integration in 1675.
The next result resembles 3.8, but now the functions are allowed to be real valued.
3.22 integration is order preserving
Suppose (X, S, μ) is a measure space and f , g : X → R are S-measurable
functions such that ∫ f dμ and ∫ g dμ are defined. Suppose also that f (x) ≤ g(x)
for all x ∈ X. Then ∫ f dμ ≤ ∫ g dμ.
Proof The cases where ∫ f dμ = ±∞ or ∫ g dμ = ±∞ are left to the reader. Thus
we assume that ∫ | f | dμ < ∞ and ∫ |g| dμ < ∞.
The additivity (3.21) and homogeneity (3.20 with c = −1) of integration imply
that
∫ ∫ ∫
g dμ − f dμ = (g − f ) dμ.
The last integral is nonnegative because g(x) − f (x) ≥ 0 for all x ∈ X.
Measure, Integration & Real Analysis, by Sheldon Axler