The one-sample Hotelling statistic has an exact F law

Given: AC, $n>p\ge 1$, an iid $N_p(\mu ,\mathrm{\Sigma })$ sample with $\mathrm{\Sigma }\succ 0$, and the null $\mu ={\mu }_0$.

AC is used in the Gaussian and Wishart constructions. The Axiom of Choice

The sample mean is independent of the centered scatter and $(n-1)S\sim W_p(\mathrm{\Sigma },n-1)$; for $n>p$, $S\succ 0$ almost surely. The Gaussian sample mean is independent of the Wishart sample scatter

The Wishart law is the law of a sum of independent Gaussian outer products, with scale $\mathrm{\Sigma }$ and integer degrees of freedom. Wishart law by Gaussian outer products

A Gaussian vector with covariance $\mathrm{\Sigma }$ is obtained by multiplying a standard normal by ${\mathrm{\Sigma }}^{1\mathrm{/}2}$. Multivariate normal law, including singular covariance

For independent $W\sim W_p(I_p,m)$ and $Z\sim N_p(0,I_p)$ with $m\ge p$, $W\succ 0$ almost surely and $\frac{m-p+1}{p}{Z}^T{W}^{-1}Z\sim F_{p,m-p+1}$. The standard Wishart inverse quadratic form has an exact Fisher law

The Fisher law has numerator and denominator degrees of freedom as given by its independent chi-square ratio definition. Student t and Fisher F laws Chi-square laws, including zero degrees

Positive-degree chi-square variables are positive almost surely, so the Fisher denominator is positive. Positive-degree chi-square laws have a positive density and regular CDF

Every positive-semidefinite symmetric matrix has a positive square root; if $\mathrm{\Sigma }\succ 0$ and ${\mathrm{\Sigma }}^{1\mathrm{/}2}v=0$, then $v^T\mathrm{\Sigma }v=\mathrm{\parallel }{\mathrm{\Sigma }}^{1\mathrm{/}2}v{\mathrm{\parallel }}^2=0$, so the root is injective and hence invertible. A non-negative operator has a unique non-negative square root

Measurable functions of independent random elements remain independent. Measurable coordinatewise functions preserve independence

The sample map $S$ is continuous by the model definition, and Sylvester's criterion expresses ${\mathcal{S}}_{++}^p$ as finitely many strict polynomial inequalities; hence $\{S\succ 0\}$ is Borel. Inversion is continuous on invertible matrices, and continuous Euclidean maps are Borel under countable choice, which follows here from AC. The iid multivariate-normal sample and covariance conventions Sylvester's criterion: a real symmetric $n\times n$ matrix with $n\ge 1$ is positive definite if and only if all leading principal minors are positive Continuous functions on Euclidean spaces are Borel measurable AC supplies countable selections and prescribed serial paths