Let \(X\) be a nonempty set, let \(E\) be a sublattice of \([0,\infty]^X\) closed under multiplication by positive scalars, and let \(I: E\to [0,\infty]\) be isotone and positive-homogeneous. Moreover, it is assumed that (a) \(0\in E\) and \(I(0)= 0\); (b) \(f\wedge t\), \((f- t)^+\in E\) and \(I(f)= I(f\wedge t)+ I((f- t)^+)\) for all \(f\in E\) and \(t> 0\); (c) \(I(f)= \sup\{I((f- a)^+\wedge (b- a)):0< a< b\}\) for all \(f\in E\). The author is mainy concerned with representing \(I\) which is, in addition, upward/downward \(\sigma\)- or \(\tau\)-continuous as an integral of Choquet type with respect to a set function defined on a lattice of subsets of \(X\) with certain outer/inner regularity and continuity properties. The principal results are farreaching generalizations and improvements of the classical Daniell-Stone and Riesz representation theorems. The upward and downward cases are discussed in a parallel manner although the latter requires \(I\) to be finite and \(E\) to consist of only finite functions. In each of these cases the treatment of \(\sigma\)- and \(\tau\)-continuity is unified. A comparison of the present approach with the traditional Daniell-Stone procedure is made by the author elsewhere [Arch. Math. 72, No. 3, 192-205 (1999; Zbl 0924.28007)]. A basic tool in the paper is an integral representation of \(I\) (without any continuity properties) due to G. H. Greco (1982), the proof of which is also included. Another basic tool is the following interesting theorem: If \(I:[0,\infty]^X\to [0,\infty]\) is sub/supermodular and satisfies (a) and (c), then it is sub/superadditive. This is a version of a result announced (but not given a complete proof) by G. Choquet (1953-54). Furthermore, the notation, terminology, ideas and results of the author's book ``Measure and integration. An advanced course in basic procedures and applications'' (1997; Zbl 0887.28001) are used extensively throughout the paper. We note that related material is contained in a recent independent paper by \textit{L. Zhou} [Trans. Am. Math. Soc. 350, No. 5, 1811-1822 (1998; Zbl 0905.28006)].