Absolute loss and conditional medians

Let $\mathsf{X}=\{u\}$, $\mathsf{Y}=\{0,4\}$ with discrete sigma-algebras, $\mathsf{A}=\mathbb{R}$ with its Borel sigma-algebra, and $\mathcal{F}$ the set of all predictors from $\mathsf{X}$ to $\mathbb{R}$. Give the joint law by $P(u,0)=P(u,4)=1\mathrm{/}2$ and define the label coordinate $Y(u,y)=y$. Use absolute loss $\mathrm{\ell }(a,y)=\mathrm{\mid }a-y\mathrm{\mid }$. A median of a probability law on $\mathbb{R}$ is a value $m$ satisfying both $P(Y\le m)\ge 1\mathrm{/}2$ and $P(Y\ge m)\ge 1\mathrm{/}2$. The conditional law of $Y$ at the only instance assigns probability $1\mathrm{/}2$ to each of $0$ and $4$, so every $m\in [0,4]$ is a median. The predictors $f_m(u)=m$ for $m\in [0,4]$ all minimize population risk, with risk $2$; every predictor with value outside $[0,4]$ has risk greater than $2$.