  Realizations of scale invariance are studied in the context of a
gravitational theory where the action (in the first order formalism) is of the
form $S = \int L_{1} \Phi d^{4}x$ + $\int L_{2}\sqrt{-g}d^{4}x$ where $\Phi$ is
a density built out of degrees of freedom, the "measure fields" independent of
$g_{\mu\nu}$ and matter fields appearing in $L_{1}$, $L_{2}$. If $L_{1}$
contains the curvature, scalar potential $V(\phi)$ and kinetic term for $\phi$,
$L_{2}$ another potential for $\phi$, $U(\phi)$, then the true vacuum state has
zero energy density, when theory is analyzed in the conformal Einstein frame
(CEF), where the equations assume the Einstein form. Global Scale invariance is
realized when $V(\phi)$ = $f_{1}e^{\alpha\phi}$ and $U(\phi)$ =
$f_{2}e^{2\alpha\phi}$. In the CEF the scalar field potential energy
$V_{eff}(\phi)$ has in, addition to a minimum at zero, a flat region for
$\alpha\phi \to\infty$, with non zero vacuum energy, which is suitable for
either a New Inflationary scenario for the Early Universe or for a slowly
rolling decaying $\Lambda$-scenario for the late universe, where the smallness
of the vacuum energy can be understood as a kind of see-saw mechanism.
