We prove that if $A$ is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if $A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $\mu$ is an arbitrary $f-$invariant measure with positive Lyapunov exponents on the boundary of $A$, then $\mu$-almost every point $q$ in the boundary of $A$ is accessible along a curve from $A$. In fact we prove the accessibility of every "good" $q$ i.e. such $q$ for which "small neighborhoods arrive at large scale" under iteration of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem on the accessibility of periodic sources.