  In this paper we introduce a reproducing kernel Hilbert space defined on
$\mathbb{R}^{d+1}$ as the tensor product of a reproducing kernel defined on the
unit sphere $\mathbb{S}^{d}$ in $\mathbb{R}^{d+1}$ and a reproducing kernel
defined on $[0,\infty)$. We extend Stolarsky's invariance principle to this
case and prove upper and lower bounds for numerical integration in the
corresponding reproducing kernel Hilbert space.
  The idea of separating the direction from the distance from the origin can
also be applied to the construction of quadrature methods. An extension of the
area-preserving Lambert transform is used to generate points on
$\mathbb{S}^{d-1}$ via lifting Sobol' points in $[0,1)^{d}$ to the sphere. The
$d$-th component of each Sobol' point, suitably transformed, provides the
distance information so that the resulting point set is normally distributed in
$\mathbb{R}^{d}$.
  Numerical tests provide evidence of the usefulness of constructing
Quasi-Monte Carlo type methods for integration in such spaces. We also test
this method on examples from financial applications (option pricing problems)
and compare the results with traditional methods for numerical integration in
$\mathbb{R}^{d}$.
