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 .