Before revision:  Denote } by L( U \times V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times V and U to the real line, respectively(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.
After revision:  Denote } by L( U \times V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times \Omega to the reals and by \mathcal{L(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| U )\colon\mathcal{L}( U \times V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant.