# johnswentworth comments on Basic Inframeasure Theory

• A positive functional for is a continuous linear function that is nonnegative everywhere on .

I got really confused by this in conjunction with proposition 1. A few points of confusion:

• The decomposition of into rather than . I’m sure this is standard somewhere, but I had to read back a ways to realize that is negative in the constraint .

• This does not match wikipedia’s definition of a positive linear functional; that only requires that the functional be positive on the positive elements of the underlying space.

• We seem to be talking about affine functions, not linear functions, but then Theorem 1 works around that by throwing in the constant .

• Yeah, looking back, I should probably fix the m- part and have the signs being consistent with the usual usage where it’s a measure minus another one, instead of the addition of two signed measures, one a measure and one a negative measure. May be a bit of a pain to fix, though, the proof pages are extremely laggy to edit.

Wikipedia’s definition can be matched up with our definition by fixing a partial order where iff there’s a that’s a sa-measure s.t. , and this generalizes to any closed convex cone. I lifted the definition of “positive functional” from Vanessa, though, you’d have to chat with her about it.

We’re talking about linear functions, not affine ones. is linear, not affine (regardless of f and c, as long as they’re in and , respectively). Observe that it maps the zero of to 0.