(Existence of Lebesgue measure).
There exists a
of subsets of
and a mapping (Lebesgue
includes all open subsets of
is a cube in
is the volume of
for any collection of subsets
is a complete measure space.
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
measure via addition of null sets