We fully characterize discrete-time dynamic convex expectations (\mathcal{E }  upper semianalytic functions - in particular we work without a reference measure and do not assume essential suprema to exist. It is shown that \mathcal{E } (\cdot|U)}\colon }\to(U) be a sublinear increasing functional which leaves L(U) invariant. We prove that there exists a set-valued mapping P_V from U to the set of probabilities on V with compact convex values and analytic graph such that E(X|U)(u)= 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 if and only if a dual representation of \mathcal{E_t in terms of conditional expectations minus the convex conjugate of \mathcal{E}_t holds true, where the conjugate is lower semianalytic with pointwise weakly compact level sets. Moreover, we provide a dual characterization of the dynamic property , i.e. we show that \mathcal{E}_t} (\cdot)}\colon } \to which leaves the constants invariant, the tower property \mathcal{E}(\cdot)} =\mathcal{E} _t\circ\mathcal{E_{t+1} if and only if the convex conjugate of \mathcal{E}_t has an additive form. We also consider dynamic convex expectations defined on the set of discrete-time stochastic processes} (\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} .
Given two Polish spaces U and V, denote by \mathcal{L \times} V) and \mathcal{L upper semianalytic functions from U \times} V and U to the real line, respectively. Let \mathcal{E(\cdot|U)}\colon\mathcal{L \times}V)\to\mathcal{L(U) be a sublinear increasing functional which leaves L(U) invariant. We prove that there exists a set-valued mapping P_V from U to the set of probabilities on V with compact convex values and analytic graph such that E(X|U)(u)= 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 _t in terms of conditional expectations minus the convex conjugate of \mathcal{E}_t holds true, where the conjugate is lower semianalytic with pointwise weakly compact level sets. Moreover, we provide a dual characterization of the dynamic property , i.e. we show that \mathcal{E}_t} . Further, given another sublinear increasing functional \mathcal{E(\cdot)}\colon\mathcal{L \times} V)\to\mathbb{R which leaves the constants invariant, the tower property \mathcal{E}(\cdot)} =\mathcal{E} _{t+1} if and only if the convex conjugate of \mathcal{E}_t has an additive form. We also consider dynamic convex expectations defined on the set of discrete-time stochastic processes} (\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} .
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