Every posterior median minimizes absolute risk when the first absolute moment is finite

The absolute loss $L(\theta ,a)=\mathrm{\mid }\theta -a\mathrm{\mid }$ is continuous on ${\mathbb{R}}^2$, hence product-Borel measurable as required by [F2]. The function $(y,\theta )\mapsto \mathrm{\mid }\theta \mathrm{\mid }$ is product-measurable, so [F4] makes its posterior integral measurable and hence $D_1$ a measurable data domain. For $y\in D_1$ and every finite $a$, $\mathrm{\mid }\theta -a\mathrm{\mid }\le \mathrm{\mid }\theta \mathrm{\mid }+\mathrm{\mid }a\mathrm{\mid }$ makes the posterior risk finite.

Fix $y\in D_1$ and a median $m$. For any $a>m$, put $d=a-m$ and partition $\mathbb{R}$ into $A=(-\mathrm{\infty },m]$, $B=(m,a)$, and $C=[a,\mathrm{\infty })$. On these three sets respectively, $\mathrm{\mid }\theta -a\mathrm{\mid }-\mathrm{\mid }\theta -m\mathrm{\mid }$ equals $d$, $a+m-2\theta$, and $-d$. On $B$, $a+m-2\theta \ge -d$. Therefore ${\rho }_y(a)-{\rho }_y(m)\ge d\{{\mathrm{\Pi }}_y(A)-{\mathrm{\Pi }}_y(B)-{\mathrm{\Pi }}_y(C)\}=d\{2{\mathrm{\Pi }}_y(A)-1\}\ge 0$, where the last inequality is the first median condition [F1].

If $a<m$, put $d=m-a$ and partition into $A=(-\mathrm{\infty },a]$, $B=(a,m)$, and $C=[m,\mathrm{\infty })$. On these sets the loss difference is respectively $-d$, $2\theta -a-m$, and $d$; on $B$, $2\theta -a-m\ge -d$. Hence ${\rho }_y(a)-{\rho }_y(m)\ge d\{-{\mathrm{\Pi }}_y(A)-{\mathrm{\Pi }}_y(B)+{\mathrm{\Pi }}_y(C)\}=d\{2{\mathrm{\Pi }}_y(C)-1\}\ge 0$ by the second median condition [F1]. Equality holds when $a=m$. Thus every median is a pointwise minimizer; [F3] supplies a measurable median rule, whose finite risk on $D_1$ and attained minimum satisfy [F2].