  Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to
approximate integrals over the unit cube with respect to the uniform measure.
In this paper we discuss QMC integration with respect to general product
measures defined on an arbitrary cube. We only require that the cumulative
distribution function is invertible. We develop a worst-case error bound and
study the dependence of the error on the number of points and the dimension for
digital nets and sequences as well as polynomial lattice point sets, which are
mapped to the domain using the inverse cumulative distribution function. We do
not require any smoothness properties of the probability density function and
the worst-case error does not depend on the particular choice of density
function and its smoothness. The component-by-component construction of
polynomial lattice rules is based on a criterion which depends only on the size
of the cube but is otherwise independent of the product measure.
