Page 141 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes Therefore, each integral on the right of (1) is finite.

In particular, g < ,
E
which shows that g is an integrable function over E.

Since f = (f g) g
E E E

f g = (f g) .
E E E

11.1.4 The General Lebesgue Integral

+
For the positive part f of a function f, we define
+
f = max (f, 0)
–
and negative part f by f –
–
f = max (–f, 0)
–
+
and that f is measurable if and only if both f and f are measurable.

Note f = f – f –
+
and |f| = f + f –
+
–
+
Definition: A measurable function f is said to be Lebesgue integrable over E if f and f are both
Lebesgue integrable over E. In this case, we define f f f .
E E E
Theorem 6: Let f and g be integrable over E, then

(a) The function of f is integrable over E, and cf c f .
E E
(b) Sum of two integrable functions is integrable i.e. the function f + g is integral over E, and

(f g) = f g
E E E

(c) If f g a.e., then f g .
E E
(d) If A and B are disjoint measurable sets contained in E, then

f = f f .
A B A B

134 LOVELY PROFESSIONAL UNIVERSITY