Near equivalence on metric spaces and a nonstandard central limit theorem

This article proves a nonstandard Central Limit Theorem (CLT) in the sense of Nelson’s Radically Elementary Probability Theory. The CLT proved here is obtained by establishing the near equivalence of standardized averages obtained from L2 IID random variables to the standardized average resulting from a binomial CLT. A nonstandard model for near equivalence on metric spaces replaces conventional results of weak convergence. Statements and proofs remain radically elementary without applying the full Internal Set Theory. A nonstandard notion of normality is discussed.