  The coordinates along any fixed direction(s), of points on the sphere
$S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$
approaches infinity. We revisit this classical result from a nonstandard
analysis perspective, providing a new proof by working with hyperfinite
dimensional spheres. We also set up a nonstandard theory for the asymptotic
behavior of integrals over varying domains in general. We obtain a new proof of
the Riemann--Lebesgue lemma as a by-product of this theory. We finally show
that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian
moment of an order larger than one, its expectation is given by a Loeb integral
integral over a hyperfinite dimensional sphere. Some useful inequalities
between high-dimensional spherical means of $f$ and its Gaussian mean are
obtained in order to complete the above proof. A review of the requisite
nonstandard analysis is provided.
