  In 1987, J. H. Elton, has proved the first fundamental result in convergence
of IFS, the Elton's Ergodic Theorem. In this work we prove the natural
extension of this theorem to the projected Hutchinson measure $\mu_{\alpha}$
associated to a GIFSpdp $\mathcal{S}=\left(X, (\phi_j:X^{m} \to X)_{j=0,1, ...,
n-1}, (p_j)_{j=0,1, ..., n-1}\right),$ in a compact metric space $(X,d)$. More
precisely, the average along of the trajectories $x_{n}(a)$ of the GIFS,
starting in any initial points $x_0, ..., x_{m-1} \in X$ satisfies, for any $f
\in C(X , \mathbb{R})$, $$\lim_{N\to +\infty} \frac{1}{N}\sum_{n=0 }^{N-1}
f(x_{n}(a)) = \int_{X} f(t) d\mu_{\alpha}(t),$$ for almost all $a \in
\Omega=\{0,1, ..., n-1\}^{\mathbb{N}}$, the symbolic space. Additionally, we
give some examples and applications to Chaos Games and Nonautonomous Dynamical
Systems defined by finite difference equations.
