<p> In a seminal paper, Choquet introduced an integral formula toextend a monotone increasing setfunction on a sigma-algebra to a (nonlinear) functionalon bounded measurable functions. The most important special case is whenthe setfunction is submodular; then this functional is convex (and vice versa). Inthe finite case, an analogous extension was introduced by this author; this is arather special case, but no monotonicity was assumed. In this note we show thatChoquet's integral formula can be applied to all submodular setfunctions, andthe resulting functional is still convex. We extend the construction to submodularsetfunctions defined on a set-algebra (rather than a sigma-algebra). The mainproperty of submodular setfunctions used in the proof is that they have boundedvariation. As a generalization of the convexity of the extension, we show that(under smoothness conditions) a "lopsided" version of Fubini's Theorem holds.</p>