Skip to Main Content
Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_13” 2011/5/22 23:35 page 62 #62
13-62 Real Analysis
for which
X
|f | dµ<.
The members of L
1
(µ) are called Lebesgue integrable functions
(with respect to µ)or summable functions.
Definition 13.6.21 If f (x) = u(x) + iv(x) where u(x) and v(x) are
real measurable functions on X and if f L
1
(µ), we define
E
fdµ =
E
u
+
dµ
E
u
dµ+i
E
v
+
dµi
E
v
dµ for every E M
where u
+
and u
are positive and negative variations of u, v
+
and
v
are positive and negative variations of v. Thus u
+
, u
, v
+
, v
are
real non-negative measurable functions. Hence the four integrals on
the above definition exist. Further
u
+
≤|u|≤|f |, u
≤|u|≤|f |
v
+
≤|v|≤|f |, v
≤|v|≤|f |
so that each of these four integrals is finite ( since f L
1
(µ) ). Thus
E
fdµ is defined as a complex ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Start your free trial

You might also like

Make: Calculus

Make: Calculus

Joan Horvath, Rich Cameron
Complex Analysis

Complex Analysis

ITL Education

Publisher Resources

ISBN: 9781299447561Publisher Website