Dominance of a Dynamical Measure and Disappearance of the Cosmological Constant

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.