  Using techniques of optimal transportation and gradient flows in metric
spaces, we extend the notion of Riemannian Curvature Dimension condition
$RCD(K,\infty)$ introduced (in case the reference measure is finite) by
Giuseppe Savare', the first and the second author, to the case the reference
measure is $\sigma$-finite; in this way the theory includes natural examples as
the euclidean $n$-dimensional space endowed with the Lebesgue measure, and
noncompact manifolds with bounded geometry endowed with the Riemannian volume
measure. Another major goal of the paper is to simplify the axiomatization of
$RCD(K,\infty)$ (even in case of finite reference measure) replacing the
assumption of strict $CD(K,\infty)$ with the classic notion of $CD(K,\infty)$.
