Student t and Fisher F laws

Assume the Axiom of Choice. For a positive integer $k$, take the product probability law of an independent standard normal $G$ and an independent $Q_k\sim {\chi }_k^2$. The Student t law with $k$ degrees of freedom, denoted $t_k$, is the law of $T_k=\{\begin{array}{ll}{\displaystyle \frac{G}{\sqrt{Q_k\mathrm{/}k}},}& Q_k>0,\\ 0,& Q_k\le 0.\end{array}$ For positive integers $r,s$, take independent $Q_r\sim {\chi }_r^2$ and $Q_s\sim {\chi }_s^2$. The Fisher F law with degrees $(r,s)$, denoted $F_{r,s}$, is the law of $F_{r,s}=\{\begin{array}{ll}{\displaystyle \frac{Q_r\mathrm{/}r}{Q_s\mathrm{/}s},}& Q_s>0,\\ 0,& Q_s\le 0.\end{array}$ The displayed piecewise maps are defined on all of ${\mathbb{R}}^2$ and are Borel measurable: each quotient is continuous on its Borel positive-denominator region, and the value on the complementary Borel nonpositive-denominator region is constant. The positive-degree chi-square lemma proves that the denominators are positive almost surely, so these versions agree almost surely with the usual quotients. Zero degrees are excluded from both definitions.