function representation similarly to Jaffray. He gives conditions for existence of a utility function representation on a probability space, that is upper semicontinuous or lower semicontinuous in the order topology. Herden defines a utility function representation of a weak preorder as an isotone function such that . Herden proved that a weak preorder on X has a continuous utility function, if and only if there exists a countable family E of separable systems on X such that, for all pairs , there is a separable system F in E, such that B is contained in all sets in F, and A is not contained in any set in F. He shows that this theorem implies Peleg's representation theorem. In a follow-up paper he clarifies the relation between this theorem and classical utility representation theorems on complete orders. Multi-utility representation A multi-utility representation (MUR) of a relation is a set U of utility functions, such that . In other words, A is preferred to B if and only if all utility functions in the set U unanimously hold this preference. The concept was introduced by Efe Ok. Every preorder (reflexive and transitive relation) has a trivial MUR. Moreover, every preorder with closed upper contour sets has an upper-semicontinuous MUR, and every preorder with closed lower contour sets has a lower-semicontinuous MUR. However, not every preorder with closed upper and lower contour sets has a continuous MUR. Ok and Evren present several conditions on the existence of a continuous MUR: has a continuous MUR if-and-only-if (X,) is a semi-normally-preordered topological space. If X is a locally compact and sigma-compact Hausdorff space, and is a closed subset of , then has a continuous MUR. This in particular holds if X is a nonempty closed subset of a Euclidean space. If X is any topological space, and is a preorder with closed upper and lower contour sets, that satisfies strong local non-satiation and an additional property called niceness, then has a continuous MUR. All the representations guaranteed by the above theorems might contain infinitely many utilities, and even uncountably many utilities. In practice, it is often important to have a finite MUR - a MUR with finitely many utilities. Evren and Ok prove there exists a finite MUR where all utilities are upper[lower] semicontinuous for any weak preference relation satisfying the following conditions: Reflexive and Transitive (that is, is a weak preorder); All upper[lower]