  Using Caratheodory measures, we associate to each positive orbit ${\mathcal
O}_{f}^{+}(x)$ of a measurable map $f$, a Borel measure $\eta_{x}$. We show
that $\eta_{x}$ is $f$-invariant whenever $f$ is continuous or $\eta_{x}$ is a
probability. These measures are used to study the \emph{historic} points of the
system, that is, \emph{points with no Birkhoff averages}, and we construct
topologically generic subset of \emph{wild historic points} for wide classes of
dynamical models. We use properties of the measure $\eta_x$ to deduce some
features of the dynamical system involved, like the \emph{existence of
heteroclinic connections from the existence of open sets of historic points}.
