Integration on the Hilbert Cube

Juan Carlos Sampedro

Published 2017-05-03Version 1

The aim of this article is to generalize the Lebesgue integration theory to $\mathbb{R}^{\mathbb{N}}$ within a preliminary measure theory, just as an extension of finite dimensional Lebesgue integral. We'll state an elementary but rigorous integration calculus on such space and we'll see that the integration on the Hilbert cube has important existence properties. The main result of this article is to prove that the space of integrable functions on the Hilbert cube is a Banach space, a fact that allow us to apply Banach space theory's results to this kind of functions. Finally, we will give some examples that show the ease of use of this theory.