Finite-area envelope for a ratio-of-uniforms region

A finite bounding rectangle exists when $\sqrt{h(x)}$ and $x\sqrt{h(x)}$ are bounded. This is not automatic for every probability density function: tails proportional to $(1+|x|{)}^{-3/2}$ give an unbounded second coordinate. In principle any finite-area measurable region has an open envelope of finite area, decomposable up to null boundaries into disjoint dyadic rectangles. Select a rectangle with probability proportional to its area, then sample its two coordinates uniformly and reject outside the target region. The accepted point is uniform on the region, with acceptance equal to the ratio of their areas. Constructing an effective envelope remains a density-specific computational step.