THE POINTWISE LIMIT SET OF METRIC INTEGRAL OPERATORS APPROXIMATING SET-VALUED FUNCTIONS

For set-valued functions (SVFs, multifunctions), mapping a compact interval [a,b] into the space of compact non-empty subsets of R^d, we study the pointwise limits of metric integral approximation operators. In our earlier papers, we have considered convergence of metric Fourier approximations and metric adaptations of some classical integral approximating operators for SVFs of bounded variation with compact graphs. While the pointwise limit of a sequence of these approximants at a point of continuity x of the set-valued function F is F(x), the limit set at a jump point is described in terms of the metric selections of the multifunction. In this paper, we show that, under certain assumptions on F, the limit set at x equals the metric average of the left and the right limits of F at x. This result extends the known classical theorems from the case of real-valued functions to SVFs.