Maps between spaces of measures on measurable spaces $(X,Sigma_X)$ and $(Y, Sigma_Y)$ are treated as generalized functions between $X$ and $Y$.
The purpose of this article is to present the construction and basic properties of the general Bochner integral. The approach presented here is based on the ideas from the book The Bochner Integral by J. Mikusinski where the integral is presented for
functions defined on $mathbb{R}^N$. In this article we present a more general and simplified construction of the Bochner integral on abstract measure spaces. An extension of the construction to functions with values in a locally convex space is also considered.
The basic properties of the Daniell integral are presented. We do not use the standard approach of introducing auxiliary spaces of the over-functions and under-functions. Instead, we use a simple and direct approach based on approximating integrable
functions by absolutely convergent series of simple functions.
A space of pseudoquotients $mathcal{B}(X,S)$ is defined as equivalence classes of pairs $(x,f)$, where $x$ is an element of a non-empty set $X$, $f$ is an element of $S$, a commutative semigroup of injective maps from $X$ to $X$, and $(x,f) sim (y,g)
$ if $gx=fy$. In this note we consider a generalization of this construction where the assumption of commutativity of $S$ by Ore type conditions. As in the commutative case, $X$ can be identified with a subset of $mathcal{B}(X,S)$ and $S$ can be extended to a group $G$ of bijections on $mathcal{B}(X,S)$. We introduce a natural topology on $mathcal{B}(X,S)$ and show that all elements of $G$ are homeomorphisms on $mathcal{B}(X,S)$.
