(Existence of Lebesgue measure).
There exists a
-algebra
of subsets of
and a mapping (Lebesgue
measure)
such that

1. The
includes all open subsets of
.

2. If
is a cube in
then
is the volume of
.

3. The
is
-additive:
for any collection of subsets
such
that

4.
is a complete measure space.

Remark

Restriction of the above Lebesgue measure to Borel sets is not complete. The
Lebesgue measure is complete on the algebra obtained from the algebra of Borel
sets by adding all sets included in sets of measure zero (see the proposition
(
Completion of
measure via addition of null sets
)-3).