  The dilaton-gravity sector of the Two Measures Field Theory (TMT)is explored
in detail in the context of cosmology. The model possesses scale invariance
which is spontaneously broken due to the intrinsic features of the TMT
dynamics. The effective model represents an explicit example of the effective
k-essence resulting from first principles without any exotic term in the
fundamental action. Depending of the choice of regions in the parameter space,
TMT exhibits different possible outputs for cosmological dynamics: a) Absence
of initial singularity of the curvature while its time derivative is singular.
This is a sort of "sudden" singularities studied by Barrow on purely kinematic
grounds. b) Power law inflation in the subsequent stage of evolution. Depending
on the region in the parameter space (but without fine tuning) the inflation
ends with a graceful exit either into the state with zero cosmological constant
(CC) or into the state driven by both a small CC and the field phi with a
quintessence-like potential. c) Possibility of resolution of the old CC
problem. From the point of view of TMT, it becomes clear why the old CC problem
cannot be solved (without fine tuning) in conventional field theories. d) TMT
enables two ways for achieving small CC without fine tuning of dimensionfull
parameters: either by a seesaw type mechanism or due to a correspondence
principle between TMT and conventional field theories (i.e theories with only
the measure of integration sqrt{-g} in the action. e) There is a wide range of
the parameters such that in the late time universe: the equation-of-state
w=p/\rho <-1; w asymptotically (as t\to\infty) approaches -1 from below; \rho
approaches a constant, the smallness of which does not require fine tuning of
dimensionfull parameters.
