Geometrically, the positive elements of a vector space with a partial order form a cone, called the positive cone. The concept makes sense even more generally but is particularly important in operator algebras.
Then the positive cone satisfies these properties:
Conversely, if is any subset of the group with these properties, then becomes an ordered group with either of these equivalent definitions:
In this way, we have a bijection between ordered group structures and positive cones in a group.
The finite measures on a given measurable space form an ordered vector space , and the positive cone consists precisely of the finite positive measures. But we often want to allow positive measures to take infinite values. The space of (possibly infinite) positive measures is the extended positive cone of .
I only know the general definition in some rather limited cases:
Let be a -algebra, and let be its predual. Recall that is the space of continuous linear maps from to the base field. The extended positive cone of is the space of lower semicontinuous linear maps from the positive cone of to the space of extended positive real numbers.
The extended positive real numbers are really showing up in their guise as the nonnegative lower real numbers (the appropriate targets for a lower semicontinuous map), and the extended positive cone is really a generalisation of the nonnegative upper reals. In particular, the extended positive cone of itself is .
This doesn't include the motivating example, but the following generalisation does:
Let be a module over the -algebra , and let have the structure of an ordered group such that the action of preserves order (in that a positive element acting on a positive element gives a positive element). Then the extended positive cone of consists of formal infinitary -linear combinations of positive elements of modulo the (hopefully) obvious equivalence relation.
When applied to the space of finite measures on a localisable measurable space (acted on by the -algebra ), this should give the positive measures on (but I need to check the details).
The extended positive cone of a -algebra is Definition 4.4 of: