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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 page6—#6
13-6 Real Analysis
Corollary 13.2.7 Let E R and >0, we can get an open set G E
such that m
(G) m
(E) +. In particular, m
is outer regular in the
sense that m
(E) = inf{m
(U )/U open, E U }.
Proof Without loss of generality, we can assume that m
(E)<
and choose a sequence {I
n
} of open intervals (by Theorem 13.2.2 (vi))
covering E such that
n=1
m
(I
n
) m
(E) + . Now G =
n=1
I
n
is an
open set containing E and
m
(G)
n=1
l(I
n
) =
n=1
m
(I
n
) m
(E) + .
The monotonicity of m
already shows that m
(E) is a lower bound for
{m
(U )/U open, E U }. By what we have proved, it is also clear
that any number greater than m
(E) is not a lower bound for this set.
The result follows.
13.2.1 Measurable Sets ...
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Publisher Resources

ISBN: 9781299447561Publisher Website