Is $xy\leq z$ or $xy\geq z$ quasi convex or quasi concave in some infinite subset of $\mathbb R_{\geq 0}^3$?

Question

Is $xy\leq z$ or $xy\geq z$ quasi convex in some infinite subset with positive measure of $\mathbb R_{\geq 0}^3$?

If not is it quasi-concave in some infinite subset with positive measure of $\mathbb R_{\geq 0}^3$?