  One of the goals of this article is to define a an unified setting adapted to
the description of means (normalized integrals or invariant means) on an
infinite product of measured spaces with infinite measure. We first remark that
some known examples coming from the theory of metric measured spaces and also
from oscillatory integrals are obtained as limits of means with respect to
finite measures. Then, we explore in a systematic way the limit of means of the
type $$ \lim \frac{1}{\mu(U_n)} \int_{U_n} f d\mu$$ where $\mu$ is a a
$\sigma-$finite Radon measure $\mu.$ In some cases, we get a linear extension
of the limit at infinity. Then, the mean value on an infinite product is
defined, first for cylindrical functions and secondly taking the uniform limit.
Finally, the mean value for the heuristic Lebesgue measure on a separable
infinite dimensional topological vector space (but principally on a Hilbert
space) is defined. This last object is shown to be invariant by translation,
scaling and restriction.
