We prove that there exists a function $f$ which reduces a given $\Pi^1_1$ subset $P$ of an internal set $X$ of an $\omega_1$-saturated nonstandard universe to the set $\mathbf{WF}$ of well-founded trees possessing properties similar to those possessed by the standard part map. We use $f$ to define the Lusin-Sierpinski index of points in $X$, and prove the basic properties of that index using the classical properties of the Lusin-Sierpinski index. An example of a $\Pi^1_1$ but not $\Sigma^1_1$ set is given.