his section contains a recipe
for building a smooth approximation to a measurable function.
The
is a subset of
.
The
is the boundary of
.
For an
we denote
,
where the
is the distance from
to the boundary
.
Definition
(Standard mollifier definition).
1.
2. For
is the "standard mollifier".
3. If
is locally integrable then we define the
"mollification"
for
.
Proposition
(Properties of mollifiers).
1.
.
2.
almost surely as
.
3.
uniformly on every compact subset of
.
4.
in
.
Definition
(Partition of unity definition). Let
be a subset of
and
be a finite or countable collection of subsets of
such
that
The collection of functions
is called a smooth partition of unity subordinated to
.
