  A construction of product measures is given for an arbitrary sequence of
measure spaces via outer measure techniques without imposing any condition on
the underlying measure spaces. This approach concludes finally the problem of
the existence of product measures in an elementary manner. Moreover, the
$L_{p}$ spaces of this measures are simplified in terms of finite product
measures following the approach of [21]. This decomposition simplifies infinite
dimensional integration and gives to this theory a computational framework.
