Signed measures have values either in or , to avoid the possibility of adding to . If is a signed measure space and is a measurable set, define
The set function and are respectively the positive, negative and total variations of .
Theorem If and are signed measures on a measurable space, there is a signed measure majorizing and and majorized by every other signed measure majorant and .
Proof If is a well-defined signed measure, that is if and are not both or both . Let be a maximal positivity set and be a maximal negativity set, for . Define
This sum defines a measure with the required properties.