  According to the Borde-Guth-Vilenkin (BGV) theorem an expanding region of
spacetime cannot be extended to the past beyond some boundary $\mathcal{B}$.
Therefore, the inflationary universe must have had some kind of beginning.
However, the BGW theorem says nothing about the boundary conditions on
$\mathcal{B}$, or even about its location. Here we present a single-scalar
field model of the Two-Measures Theory, where the non-Riemannian volume element
$\Upsilon d^4x$ is present in the action. As a result of the model dynamics, an
upper bound $\varphi_0$ of admissible values of the scalar field $\varphi$
appears, which sets the position of $\mathcal{B}$ in the form of a spacelike
hypersurface $\Upsilon(x)=0$ with a boundary condition: $\Upsilon\rightarrow
0^+$ as $\varphi\rightarrow\varphi_0^{\,\,\,-}$. A detailed study has
established that if the initial kinetic energy density $\rho_{kin}^{(in)}$
prevails over initial gradient energy density $\rho_{grad}^{(in)}$ then there
is an interval of initial values $\varphi_{in}^{(min)}\leq \varphi_{in}
<\varphi_0$, where $\rho_{kin}^{(in)}$ and $\rho_{grad}^{(in)}$ cannot exceed
the potential energy density and hence the initial conditions necessary for the
onset of inflation are satisfied. It is shown that under almost all possible
left-handed boundary conditions on $\mathcal{B}$, that is where
$\Upsilon\rightarrow 0^-$, the metric tensor in the Einstein frame has a jump
discontinuity on $\mathcal{B}$, so the Christoffel connection coefficients are
not defined on the spacelike hypersurface $\Upsilon=0$. Thus, if
$\varphi_{in}^{(min)}\leq \varphi_{in}<\varphi_0$ and
$\rho_{kin}^{(in)}>\rho_{grad}^{(in)}$, then there was an inflationary stage in
the history of our Universe and the congruence of timelike geodesics cannot be
extended to the past beyond the hypersurface $\Upsilon=0$.
