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
.