  In this report, we revisit the work of Pilleboue et al. [2015], providing a
representation-theoretic derivation of the closed-form expression for the
expected value and variance in homogeneous Monte Carlo integration. We show
that the results obtained for the variance estimation of Monte Carlo
integration on the torus, the sphere, and Euclidean space can be formulated as
specific instances of a more general theory. We review the related
representation theory and show how it can be used to derive a closed-form
solution.
