A theory of integration of
compact set-valued functions is provided by applying the McShane φ-integral. This
integral is a Riemann-type integral and includes the Bochner, Lebesgue and other
types of integrals, and by using Riemann sums it avoids deep measure theory. Thus,
the φ-integral of set-valued functions contains other types of integrals such as the
Hukuhara and Debreu integrals. Generalizations of known results, including the
convexity of the integral, are obtained, and the techniques do not require measure
theory. Further, if a set-valued function is φ-integrable, then its integral equals the
Aumann integral, where the latter is defined as the collection of integrals of
selections.
