  The dynamically defined measure (DDM) $\Phi$ arising from a finite measure
$\phi_0$ on an initial $\sigma$-algebra on a set and an invertible map acting
on the latter is considered. Several lower bounds for it are obtained and
sufficient conditions for its positivity are deduced under the general
assumption that there exists an invariant measure $\Lambda$ such that
$\Lambda\ll\phi_0$.
  In particular, DDMs arising from the Hellinger integral
$\mathcal{J}_\alpha(\Lambda,\phi_0)\geq\mathcal{H}^{\alpha,0}(\Lambda,\phi_0)\geq\mathcal{H}_\alpha(\Lambda,\phi_0)$
are constructed with $\mathcal{H}_{0}\left(\Lambda,\phi_0\right)(Q) = \Phi(Q)$,
$\mathcal{H}_{1}\left(\Lambda,\phi_0\right)(Q) = \Lambda(Q)$, and
\[\Phi(Q)^{1-\alpha}\Lambda(Q)^{\alpha}\geq\mathcal{J}_{\alpha}\left(\Lambda,\phi_0\right)(Q)\]
for all measurable $Q$ and $\alpha\in[0,1]$, and further computable lower
bounds for them are obtained and analyzed. The function
$(0,\gamma]\owns\alpha\longmapsto\mathcal{H}_{\alpha}(\Lambda,\phi_0)$ is
computed explicitly for $\gamma\geq 1$ such that
$\int(d\Lambda/d\phi_0)^{\gamma-1}d\Lambda<\infty$ in the case of a discrete
ergodic decomposition of $\Lambda$, and the other two functions are computed
under the additional condition of the equivalence of $\phi_0$ and $\Lambda$. In
particular, if $\Lambda$ is ergodic, it is shown that the first function is
completely determined by the $\Lambda$-essential supremum (infimum) of
$d\Lambda/d\phi_0$ for all $0<\alpha<1$ ($1<\alpha\leq\gamma$), and, if it is
continuous at $0$, the above inequalities become equalities. The computation of
it enables an explicit computation of some DDMs arising as outer measure
approximations with respect to it, which demonstrates that this technique
allows to obtain new measures, and that such measures can have phase
transitions with respect to the DDM specifying the covering sets.
