Slice Sampling asks to draw from $f^{-1}]y,+\infty[$

Slice Sampling asks to draw uniformly from $f^{-1}]y,+\infty[$. Wikipedia page

However, how can we be sure that a uniform defined over the set $f^{-1}]y,+\infty[$ is in fact proper?

If I had to guess, I would say this set must be compact... is this intuition correct?

• +1. We might suppose $f$ is assumed to be at least piecewise continuous. But even if $f$ is infinitely differentiable, the sets $f^{-1}\left(y,\infty\right)$ need not be compact. Compactness isn't essential--but having finite Lebesgue measure is. The Wikipedia language about unimodality provides some clues concerning the additional implicit assumptions being made. – whuber Aug 2 '18 at 14:02

The slice sampler for a target proportional to $f$ goes by
1. simulating $u^{t+1}\sim\mathbb{I}_{(0,f(x^t))}(u)$
2. simulating $x^{t+1}\sim\mathbb{I}_{\{x;\,f(x)>u^{t+1}\}}(x)$
and both sets have finite measures, since $f(x^t)<\infty$ and $$\mu(\{x;\,f(x)>u^{t+1}\})=\int_{\{x;\,f(x)>u^{t+1}\}}\text{d}\mu(x)<\int_{ \{x;\,f(x)>u^{t+1}\} }\dfrac{f(x)}{u^{t+1}}\text{d}\mu(x)\le \frac{1}{u^{t+1}}$$ hence the Uniform distributions are well-defined.