Additivity and indefinite integration for McShane’s $P$-integral

Abstract:

Suppose $[a,b]$ is a closed real number interval, $A = \{ (p,q];(p,q] \subset (a,b]\}$, and $U$ is a real valued function on $[a,b] \times A$. For $c \in (a,b)$, necessary and sufficient conditions are given for $P$-integrability of $U$ on $(a,b]$ and $(c,b]$ to imply $P$-integrability on $(a,b]$. Suppose $U$ is $P$-integrable on $(a,b]$ and $g(x) = P\smallint _\alpha ^xU$ for each $x \in (a,b]$. Necessary and sufficient conditions are given for $g$ to be respectively continuous, bounded, and of bounded variation.