  In numerical integration, cubature methods are effective, especially when the
integrands can be well-approximated by known test functions, such as
polynomials. However, the construction of cubature formulas has not generally
been known, and existing examples only represent the particular domains of
integrands, such as hypercubes and spheres. In this study, we show that
cubature formulas can be constructed for probability measures provided that we
have an i.i.d. sampler from the measure and the mean values of given test
functions. Moreover, the proposed method also works as a means of data
compression, even if sufficient prior information of the measure is not
available.
