Measure, Integration & Real Analysis, 2021a

Section 3A Integration with Respect to a Measure 75
Suppose (X, S, μ) is a measure space and f : X → [0, ∞] is an S-measurable
function. Each S-partition A 1 ,...,A m of X leads to an approximation of f from
below by the S-measurable simple function ∑ m (
j=1 inf f ) χ
A Aj
. This suggests that
j
m
∑ μ(A j ) inf f
j=1
A j
should be an approximation from below of our intuitive notion of ∫ f dμ. Taking the
supremum of these approximations leads to our definition of ∫ f dμ.
The following result gives our first example of evaluating an integral.
3.4 integral of a characteristic function
Suppose (X, S, μ) is a measure space and E ∈S. Then
∫
χ E
dμ = μ(E).
Proof If P is the S-partition of X consisting
of E and its complement X \ E,
then clearly L(χ E
, P) = μ(E). Thus
∫
χE dμ ≥ μ(E).
To prove the inequality in the other
direction, suppose P is an S-partition
A 1 ,...,A m of X. Then μ(A j ) inf χ
A E j
equals μ(A j ) if A j ⊂ E and equals 0
otherwise. Thus
L(χ E
, P) =
= μ
∑
{j : A j ⊂E}
( ⋃
≤ μ(E).
{j : A j ⊂E}
μ(A j )
A j
)
The symbol d in the expression
∫ f dμ has no independent meaning,
but it often usefully separates f from
μ. Because the d in ∫ f dμ does not
represent another object, some
mathematicians prefer typesetting
an upright d in this situation,
producing ∫ f dμ. However, the
upright d looks jarring to some
readers who are accustomed to
italicized symbols. This book takes
the compromise position of using
slanted d instead of math-mode
italicized d in integrals.
Thus ∫ χ E
dμ ≤ μ(E), completing the proof.
3.5 Example integrals of χ Q
and χ [0, 1] \ Q
Suppose λ is Lebesgue measure on R. As a special case of the result above, we
have ∫ χ Q
dλ = 0 (because |Q| = 0). Recall that χ Q
is not Riemann integrable on
[0, 1]. Thus even at this early stage in our development of integration with respect to
a measure, we have fixed one of the deficiencies of Riemann integration.
Note also that 3.4 implies that ∫ χ [0, 1] \ Q
dλ = 1 (because |[0, 1] \ Q| = 1),
which is what we want. In contrast, the lower Riemann integral of χ [0, 1] \ Q
on [0, 1]
equals 0, which is not what we want.
Measure, Integration & Real Analysis, by Sheldon Axler