“real: chapter_13” — 2011/5/22 — 23:35 — page 37 — #37
Lebesgue Measure and Integration 13-37
13.5 INTEGRATION OF REAL-VALUED FUNCTIONS
In this section, we shall define the integral of a real-valued measur-
able functions (not necessarily non-negative) defined on a measurable
subset of
R and study its properties.
Definition 13.5.1 Let f : E →
R be a measurable function where E
is a measurable subset of
R. We define
E
f (x)dm(x) =
E
f
+
(x)dm(x) −
E
f
−
(x)dm(x)
where f
+
(x) and f
−
(x) are the positive and negative variations of f ,
respectively.
This integral exists as an extended real number only if either
E
f
+
(x)dm(x)<∞ or
E
f
−
(x)dm(x)<∞.
One sufficient condition under which
E
f (x)dm(x) exists as a real num-
ber is to assume that
E
f
+
(x)dm(x)<∞ and
E
f
−
(x)dm(x)<∞.I ...