Let $\Omega$ be a Polish space with Borel $\sigma$-field $\mathcal{ F}$ and countably generated sub $\sigma$-field $\mathcal{ G}\subset\mathcal{ F}$. Denote by $\mathcal{ L}(\mathcal{ F})$ the set of all bounded $\mathcal{ F}$-upper semianalytic functions from $\Omega$ to the reals and by $\mathcal{ L}(\mathcal{ G})$ the subset of $\mathcal{ G}$-upper semianalytic functions. Let $\mathcal{ E}(\cdot|\mathcal{ G})\colon\mathcal{ L}(\mathcal{ F})\to\mathcal{ L}(\mathcal{ G})$ be a sublinear increasing functional which leaves $\mathcal{ L}(\mathcal{ G})$ invariant. It is shown that there exists a $\mathcal{ G}$-analytic 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 $\mathcal{ E}(\cdot)\colon\mathcal{ L}(\mathcal{ F})\to\mathbb{ R}$ 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.
Categories: math.PR q-fin.RM