  The dilaton-gravity sector of the Two Measures Field Theory (TMT) is explored
in detail in the context of cosmology. The dilaton \phi dependence of the
effective Lagrangian appears only as a result of the spontaneous breakdown of
the scale invariance. If no fine tuning is made, the effective \phi-Lagrangian
p(\phi,X) depends quadratically upon the kinetic energy X. Hence TMT may
represent an explicit example of the effective k-essence resulting from first
principles without any exotic term in the fundamental action intended for
obtaining this result. Depending of the choice of regions in the parameter
space, TMT exhibits different possible outputs for cosmological dynamics: a)
Possibility of resolution of the old cosmological constant (CC) problem. From
the point of view of TMT, it becomes clear why the old CC problem cannot be
solved (without fine tuning) in the conventional field theories (i.e theories
with only the measure of integration \sqrt{-g} in the action). b) The power law
inflation without any fine tuning can end with damped oscillations of \phi
around the state with zero CC. d) There is a broad range of the parameters such
that: in the late time universe w=p/\rho <-1 and asymptotically (as t\to\infty)
approaches -1 from below; \rho approaches a cosmological constant. The
smallness of the CC may be achieved without fine tuning of dimensionfull
parameters: either by a seesaw type mechanism or due to a correspondence
principle between TMT and conventional field theories.
