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 11 #11
Lebesgue Measure and Integration 13-11
and call m as the Lebesgue measure on R. Thus the Lebesgue measure
is associated with the pair (
R, M). Further m satisfies the following
properties:
(i) ,
R M and m() = 0, m(R) =∞. For E M, m(E) 0
and
(ii) m is countably additive on
M.
For these reasons, we say that the triplet (
R, M, m) is a measure space
and m is the Lebesgue measure on
R.
Theorem 13.2.19 For any subset A of
R there exists a measurable
set E (i.e. E
M) such that A E and m
(A) = m(E).
Proof Given =
1
n
> 0 using Corollary 13.2.7 we have a sequence of
open sets {U
n
} such that U
n
A, m
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