Pseudo-Lp space and convergence

In the framework of the pseudo-analysis the classical Lp space is generalized and there are proved important properties of introduced space. Three types of convergence of sequences of measurable functions are considered in this space. The inequalities for integrals based on pseudo-integral have been recently proposed as the Hölder, Minkowski and Markov inequalities, which are applied in observing relationship among introduced convergences.