The Jordan Decomposition of an Indefinite Integral
So, after all our setup it shouldn’t be surprising that we take an integrable function and define its indefinite integral:
Now, as we’ve pointed out, this will be a measure so long as is a.e. non-negative. But now if
is any integrable function at all, the indefinite integral
is a finite signed measure.
Let’s give a Hahn decomposition corresponding to . I say that if we set
then is positive,
is negative, and
is a Hahn decomposition. Indeed, we know that
and
are measurable. Thus if
is measurable, then
is measurable, and we find
since everywhere. Similarly, we verify that
is negative.
Now we can use this to find the Jordan decomposition of . We define
That is, the upper variation of is the indefinite integral of the positive part of
, while the lower variation of
is the indefinite integral of the negative part of
. And then we can calculate
The total variation of is the indefinite integral of the absolute value of
.