Sorry, that footnote is just flat wrong, the order actually doesn’t matter here. Good catch!
There is a related thing which might work, namely taking the downwards closure of the affine subspace w.r.t. some cone which is somewhat larger than the cone of measures. For example, if your underlying space has a metric, you might consider the cone of signed measures which have non-negative integral with all positive functions whose logarithm is 1-Lipschitz.
Sorry, that footnote is just flat wrong, the order actually doesn’t matter here. Good catch!
There is a related thing which might work, namely taking the downwards closure of the affine subspace w.r.t. some cone which is somewhat larger than the cone of measures. For example, if your underlying space has a metric, you might consider the cone of signed measures which have non-negative integral with all positive functions whose logarithm is 1-Lipschitz.