Measurability theorem
In mathematics, the Hahn decomposition theorem, named after the Austrian mathematician Hans Hahn, states that for any measurable space
and any signed measure
defined on the
-algebra
, there exist two
-measurable sets,
and
, of
such that:
and
.
- For every
such that
, one has
, i.e.,
is a positive set for
.
- For every
such that
, one has
, i.e.,
is a negative set for
.
Moreover, this decomposition is essentially unique, meaning that for any other pair
of
-measurable subsets of
fulfilling the three conditions above, the symmetric differences
and
are
-null sets in the strong sense that every
-measurable subset of them has zero measure. The pair
is then called a Hahn decomposition of the signed measure
.