A positive functional for Msa(X) is a continuous linear function M±(X)⊕R→R that is nonnegative everywhere on Msa(X).

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

The decomposition of m into m++m− rather than m+−m−. I’m sure this is standard somewhere, but I had to read back a ways to realize that m−(1) is negative in the constraint b+m−(1)≥0.

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 c.

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 (m′,b′)≥(m,b) iff there’s a (m∗,b∗) that’s a sa-measure s.t. (m,b)+(m∗,b∗)=(m′,b′), 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.(m,b)↦c(m(f)+b) is linear, not affine (regardless of f and c, as long as they’re in C(X) and R, respectively). Observe that it maps the zero of M±(X)⊕R to 0.

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

The decomposition of m into m++m− rather than m+−m−. I’m sure this is standard somewhere, but I had to read back a ways to realize that m−(1) is

negativein the constraint b+m−(1)≥0.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 c.

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 (m′,b′)≥(m,b) iff there’s a (m∗,b∗) that’s a sa-measure s.t. (m,b)+(m∗,b∗)=(m′,b′), 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.(m,b)↦c(m(f)+b) is linear, not affine (regardless of f and c, as long as they’re in C(X) and R, respectively). Observe that it maps the zero of M±(X)⊕R to 0.