In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval $\left[a,b\right]$ into a Banach space $X.$ It is shown that a Denjoy-Bochner integrable function on $\left[a,b\right]$ is Denjoy-Riemann integrable on $\left[a,b\right]$, that a Denjoy-Riemann integrable function on $\left[a,b\right]$ is Denjoy-McShane integrable on $\left[a,b\right]$ and that a Denjoy-McShane integrable function on $\left[a,b\right]$ is Denjoy-Pettis integrable on $\left[a,b\right].$ In addition, it is shown that for spaces that do not contain a copy of $c_0$, a measurable Denjoy-McShane integrable...

The assertion in question comes from the short final section in Theory of capacities of Choquet (1953/54), in connection with his prototype of the subsequent Choquet integral. The problem was whether and when this operation is additive. Choquet had the much more abstract idea that all functionals in a certain wide class must be subadditive, and similarly for superadditivity. His treatment of this point was more like an outline, and his proof limited to a rather narrow special case. Thus the proper...

A definition of “Šipoš integral” is given, similarly to [3],[5],[10], for real-valued functions and with respect to Dedekind complete Riesz-space-valued “capacities”. A comparison of Choquet and Šipoš-type integrals is given, and some fundamental properties and some convergence theorems for the Šipoš integral are proved.