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.
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.