  Multidimensional integration by parts formulas are valid under the standard
assumption that one of the functions is continuous and the other has bounded
Hardy-Krause variation. Motivated by several recently developed results in the
probabilistic context of price and risk bounds, we provide a version of an
integration by parts formula for Lebesgue integrals on a possibly unbounded
domain where a componentwise left-/right-continuous and measure inducing
function is integrated with respect to the signed measure induced by a measure
inducing, componentwise right-/left-continuous, bounded, and grounded function.
For this purpose, we extensively analyze measure inducing functions and
characterize them in terms of Delta-monotone functions. We provide, as a
consequence of the integration by parts formula, several convergence results
which allow an extension of the Lebesgue integral of a measure inducing
function to the case that one integrates with respect to a Lipschitz-continuous
semi-copula.
