Maps to Spaces of Compacta Determined by Limit Sets

For a sequence of functions on the unit disk $D\subset\mathbb C$, the map of the boundary circle to a space of compact sets with Hausdorff metric which takes each point $e^{i\theta}\in\partial D$ to the limit set of the sequence of functions at this point is considered. It is shown that such a map is of Borel class at most 4.