  We consider an action which consists of two terms: the first S_{1}=\int
L_{1}\Phi d^{4}x and the second S_{2}=\int L_{2}\sqrt{-g}d^{4}x where \Phi is a
measure which has to be determined dynamically. S_{1} satisfies the requirement
that the transformation L_{1}\to L_{1}+const does not effect equations of
motion. In the first order formalism, a constraint appears which allows to
solve \chi =\Phi/\sqrt{-g}. Then, in a true vacuum state (TVS), \chi\to\infty
and in the conformal Einstein frame no singularities are present, the energy
density of TVS is zero without fine tuning of any scalar potential in S_{1} or
S_{2}. When considering only a linear potential for a scalar field \phi in
S_{1}, the continuous symmetry \phi\to\phi+const is respected. Surprisingly, in
this case SSB takes place while no massless ("Goldstone") boson appears.
