SAMS Subject Classification Number: 30

The most important and common features and resemblances in the different approaches to integrals are gathered to obtain a more unified and simpler system of properties suitable for the definition of integration of functions taking values in topological linear spaces with respect to vector-valued integrators. As examples of application, we show that such a novel approach is flexible enough to give a further generalization of the spectral theorem on Hilbert spaces and that such an approach can also be used to push stochastic integration to the more general setting of integrable processes taking values in topological vector spaces.