The Bochner and the monotone integrals with respect to a nuclear finitely additive measure

The main idea of this paper is to use the representation of a complete nuclear space /E /as a projective limit of a family of Hilbert spaces in order to compare different concepts of integration with respect to an /E/-valued countably and finitely additive measure. Indeed in a previous paper we have already described the use of scattered dilation for the case of Hilbert valued finitely additive measures. Here the same comparison is carried out by showing that in the countably additive case the global integration in both senses is equivalent to the integration with respect to each projection. Next we define suitably orthogonally scattered dilation, and apply them in both the countably and the finitely additive case.