  We study field theory models in the context of a gravitational theory based
on the requirement that the measure of integration in the action is not
necessarily \sqrt{-g} but it is determined dynamically through additional
degrees of freedom, like four scalar fields \phi_{a}. We study three
possibilities for the general structure of the theory: (A) The total action has
the form S=\int\Phi Ld^{4}x where the measure \Phi is built from the scalars
\phi_{a} in such a way that the transformation L\to L+const does not effect
equations of motion. Then an infinite dimensional shifts group of the measure
fields (SGMF) \phi_{a} by arbitrary functions of the Lagrangian density L is a
symmetry group of the action. (B) The total action has the form S=S_{1}+S_{2},
S_{1}=\int\Phi L_{1}d^{4}x, S_{2}=\int\sqrt{-g}L_{2}d^{4}x which is the only
model different from (A) and invariant under SGMF (but now with f_{a}=
f_{a}(L_{1})). Similarly, now only S_{1} satisfies the requirement that the
transformation L_{1}\to L_{1}+const does not effect equations of motion. Both
in the case (A) and in the case (B) it is assumed that L, L_{1}, L_{2} do not
depend on \phi_{a}. (C) The action includes a term which breaks the SGMF
symmetry. It is shown that in the first order formalism in cases (A) and (B)
the CCP is solved: the effective potential vanishes in a true vacuum state
(TVS) without fine tuning. In the case (C), the breaking of the SGMF symmetry
induces a nonzero energy density for the TVS.
