Given two Polish spaces U and V, denote and countably generated sub \sigma-field G\subsetF. Denote } by L( U \times%DIFDELCMD < } %%% V) and \mathcal{L the set of all bounded upper semianalytic functions from U \times%DIFDELCMD < } %%% 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%DIFDELCMD < }%%% V )\to\mathcal{L}( U ) be a sublinear increasing functional which leaves \mathcal{L}( U ) invariant. We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional \mathcal{E}(\cdot)\colon\mathcal{L}( U \times%DIFDELCMD < } %%% V )\to\mathbb{R} which leaves the constants invariant, the tower property \mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot| U )) is characterized via a pasting property of the representing sets of probabilities . As applications, we characterize under which conditions the product of a set of probabilities and a set of kernels is compact , and under which conditions a nonlinear version of Fubini's theorem holds true .
Let \Omega be a Polish space with Borel \sigma-field \mathcal{F and countably generated sub \sigma-field G\subsetF. Denote } by L( %DIFDELCMD < } %%% \mathcal{F the set of all bounded \mathcal{F semianalytic functions from %DIFDELCMD < } %%% \Omega to the reals and by \mathcal{L(G) the subset of \mathcal{G}-upper semianalytic functions} . Let \mathcal{E}(\cdot| \mathcal{G )\colon\mathcal{L}( %DIFDELCMD < }%%% \mathcal{F )\to\mathcal{L}( \mathcal{G ) be a sublinear increasing functional which leaves \mathcal{L}( \mathcal{G ) invariant. It is shown that there exists a \mathcal{G set-valued mapping \mathcal{P} _{\mathcal{G} from \Omega } to the set of probabilities which are concentrated on atoms of \mathcal{G with compact convex values such that \mathcal{E}(X| \mathcal{G)(\omega} )= \sup_{P\in\mathcal{P} _{\mathcal{G ( \omega )} E_P[X] if and only if \mathcal{E}(\cdot | \mathcal{G ) is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional \mathcal{E}(\cdot)\colon\mathcal{L}( %DIFDELCMD < } %%% \mathcal{F )\to\mathbb{R} which leaves the constants invariant, the tower property \mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot| \mathcal{G )) is characterized via a pasting property of the representing sets of probabilities , and the importance of analytic functions is explained. Finally, it is characterized when a nonlinear version of Fubini's theorem holds true and when the product of a set of probabilities and a set of kernels is compact .
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