Preface to the Third Edition
In
this edition several topics are added. Since these topics
do
not
fit in single sections they
are presented as appendices. They are:
[A]
Digital Expansions
of
Real Numbers
[B] Measurability
of
Limits and Derivatives
[C]
Lipschitz Condition
and
Bounded Derivative
[D]
Uniform Integrability
[E] Product-measurability
and
Factor-measurability
[F] Functions
of
Bounded Oscillation
In
[B],
we
show that
if
the limit
of
a real-valued function
on
R exists then
it
is Borel-
measurable.
In
particular
if
a real-valued function
is
differentiable then the derivative
is
Borel-measurable.
In
[C],
we
show that
if
a real-valued function satisfies a Lipschitz condition
on
[a,
b]
c R
then
it
is
differentiable
a.
e.
on
[a, b] and moreover the derivative
is
bounded
on
[a, b ].
In
[D],
we
discuss uniform integrability and equi-integrability.
In
[F],
we
define the notion
of
bounded oscillation
for
a real-valued function whose domain
of
definition
is
a closed
box
in
Rn.
We show that
for
the
particular case n = 1 a function
is
of
bounded oscillation
if
and only
if
it
is
a function
of
bounded variation.
Also
93 problems have been added. There is
now
a total
of
394 problems.
Corona del Mar, California
August,
2013
xix
J.
Yeh
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