Correction to: Measures under the flat norm as ordered normed vector space

1 Correction to: Positivity DOI 10.1007/s11117-017-0503-z

Corollary 4.16 needs to be restated as

Corollary 4.16

$\mathcal{M}_+^s(S)$ is a cone of $\mathcal{M}(S)$.

$\mathcal{M}_+^t(S)$ is a cone of $\mathcal{M}(S)$ if S is complete, because then $\mathcal{M}_+^t(S) = M_+^s(S)$. In general, $\mathcal{M}_+^t(S)$ is not a cone of $\mathcal{M}(S)$ because it is not closed.

It follows from Remark 4.20 that $\mathcal{M}_+^t(S)$ is dense in $\mathcal{M}_+^s(S)$. Assume that $\mathcal{M}_+^t(S)$ is closed and S is separable. Then, $\mathcal{M}_+^t(S) = \mathcal{M}_+(S)$. This implies that S is universally measurable; however, not all separable metric spaces are universally measurable ([1, Sect. 11.5]).