harfe comments on Basic Inframeasure Theory

[harfe]

I think Proposition 1 is false as stated because the resulting functional $f^{+}:M^{\pm }\left(X\right)\oplus R\to R$ is not always continuous (wrt the KR-metric). The function $f:[0,1]\to [0,1]$, $x\mapsto x^{1/3}$ with $X=[0,1]$ should be a counterexample. However, the non-continuous functional $f^{+}$ should still be continuous on the set of sa-measures.

Another thing: the space of measures $M^{\pm }\left(X\right)$ is claimed to be a Banach space with the KR-norm (in the notation section). Afaik this is not true, while the space is a Banach space with the TV-norm, with the KR-metric/​norm it should not be complete and is merely a normed vector space. Also the claim (in “Basic concepts”) that $M^{\pm }\left(X\right)$ is the dual space of $C\left(X\right)$ is only true if equipped with TV-norm, not with KR-metric.

Another nitpick: in Theorem 5, the type of $h$ in the assumption is probably meant to be $C\left(X\right)\to R\cup \{-\infty \}$, instead of $C\left(X\right)\to R$.