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Abstract. This paper proves the existence of solutions to the initial value problemwhere f : [0, 1] × R M → R M may be discontinuous but is assumed to satisfy conditions of superposition-measurability, quasimonotonicity, quasisemicontinuity, and integrability. The set M can be arbitrarily large (finite or infinite); our theorem is new even for card(M ) = 2. The proof is based partly on measure-theoretic techniques used in one dimension under slightly stronger hypotheses by Rzymowski and Walachowski. Further generalizations are mentioned at the end of the paper.

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