  Given a finite-to-one factor map $\pi: (X, T) \to (Y, S)$ between topological
dynamical systems, we look into the pushforward map $\pi_*: M(X, T) \to M(Y,T)$
between sets of invariant measures. We investigate the structure of the measure
fiber $\pi_*^{-1}(\nu)$ for an arbitrary ergodic measure $\nu$ on the factor
system $Y$. We define the degree $d_{\pi,\nu}$ of the factor map $\pi$ relative
to $\nu$ and the multiplicity of each ergodic measure $\mu$ on $X$ that
projects to $\nu$, and show that the number of ergodic pre-images of $\nu$ is
$d_{\pi,\nu}$ counting multiplicity. In other words, the degree $d_{\pi,\nu}$
is the sum of the multiplicity of $\mu$ where $\mu$ runs over the ergodic
measures in the measure fiber $\pi^{-1}_*(\nu)$. This generalizes the following
folklore result in symbolic dynamics for lifting fully supported invariant
measures: Given a finite-to-one factor code $\pi: X \to Y$ between irreducible
sofic shifts and an ergodic measure $\nu$ on $Y$ with full support,
$\pi^{-1}_*(\nu)$ has at most $d_\pi$ ergodic measures in it, where $d_\pi$ is
the degree of $\pi$. We apply our theory of structure of measure fibers to the
special case of symbolic dynamical systems. In this case, we demonstrate that
one can list all (finitely many) ergodic measures in the measure fiber
$\pi^{-1}_*(\nu)$.
