  In a previous paper the authors developed a H^1-BMO theory for unbounded
metric measure spaces $(M,\rho,m)$ of infinite measure that are locally
doubling and satisfy two geometric properties, called "approximate midpoint"
property and "isoperimetric" property. In this paper we develop a similar
theory for spaces of finite measure. We prove that all the results that hold in
the infinite measure case have their counterparts in the finite measure case.
Finally, we show that the theory applies to a class of unbounded, complete
Riemannian manifolds of finite measure and to a class of metric measure spaces
of the form (R^n,\rho_\phi, m_\phi), where dm_\phi=e^{-\phi} dx and \rho_\phi$
is the Riemannian metric corresponding to the length element ds^2=(1+
|grad\phi|)^2(dx_1^2+...+dx_n^2)$. This generalizes previous work of the last
two authors for the Gauss space.
