Carathéodory CM-selectors for oppositely semicontinuous multifunctions of two variables

We present a new joint selection theorem on the so-called Caratheodory CM-selections for oppositely semicontinuous (in x) multifunctions F and G of variables (t, x) [belongs to] T x X where T is a complete measurable space and X is a Polish space. This theorem unifies the known theorem on Caratheodory selections for the lower Caratheodory multifunction G and a general theorem on Caratheodory epsilon-approximate selections for the H-upper Caratheodory multifunction F. The latter theorems are "T-random" versions of Michael's theorem and Cellina's theorem, respectively.