Multidimensional integration by parts formulas apply under the standard assumption that one of the functions is continuous and the other has bounded Hardy-Krause variation. Motivated by recently developed results in the probabilistic context of price and risk bounds, this paper provides a version of an integration by parts formula for the Lebesgue integral of measure-inducing functions which may both be discontinuous and may have infinite Hardy-Krause variation. To this end, we give a general definition of measure-inducing functions and establish various of their properties, such as a characterization in terms of Delta-monotone functions. As a consequence of the integration by parts formula, several convergence results are provided, allowing an extension of the Lebesgue integral of a measure-inducing function to the case where one integrates with respect to a continuous semi-copula. The latter class of aggregation functions includes quasi-copulas which serve as bounds for the dependence structure in many applications.