Here we give proofs for two versions of Lusin’s Theorem, one from Exercise 44, Ch2 in Folland’s Real Analysis and the other from the textbook used for my first year undergraduate mathematical analysis course in Beijing. The latter version is a stronger result which in addition discusses the condition for a real-valued function defined on a subset of
to be extended to the whole of
. A more general result in topology is the Tietze Extension Theorem.
See the full post here: Lusin’s Theorem and Continuous Extension
Here we let
denote the Lebesgue measure on
.
Lusin’s Theorem (Version 1)[Exercise 2.44, Folland]. Suppose
is Lebesgue measurable,
is Lebesgue measurable and
, there is a compact set
such that
and
is continuous.
Lusin’s Theorem(Version 2)[Huan]. Suppose
is Lebesgue measurable and
is a Lebesgue measurable extended real valued function with
, then
,
such that
, where
denotes the space of continuous function on
.
Continuous Extension Theorem[Huan]. Suppose
, then
can be extended to a continuous function on
if and only if
can be extended to a continuous function on the closure
of
.
Tietze Extension Theorem. Let
be normal and
be closed and let
be continuous. Then there is a map
such that
for all
. (Note that in topology, by a map we mean a continuous function. )