(Extension of measure to
sigma algebra) Under the conditions of the definition
(
Induced outer measure
) let
be the restriction of the outer measure
to the algebra
of
-measurable
sets. Then

1.
is a measure space.

2. if
is finite
(
-finite
) then so is
.

3. if
is
-finite
then
is the only extension of
to the smallest
-algebra
containing
.