Before revision:  Further, given another sublinear increasing functional \mathcal{E}(\cdot)\colon\mathcal{L}( U \times 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 .
After revision:  Further, given another sublinear increasing functional \mathcal{E}(\cdot)\colon\mathcal{L}( U \times 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 , 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 , and under which conditions a nonlinear version of Fubini's theorem holds true .