Pointwise squared-error risk for regression

Assume AC as used to select the regression version. Fix a statistical experiment indexed by $\theta$, a point $x$, a real regression target $m_{\theta }(x)$, and a real-valued estimator $T_x$ of that target, meaning a measurable statistic with target space $\mathbb{R}$ as in Statistics and estimators. At this point, use the squared-error loss $L_x(\theta ,a)=(a-m_{\theta }(x){)}^2,a\in \mathbb{R}\mathrm{.}$ Its pointwise squared-error risk is the possibly infinite quantity $R_x(\theta ,T_x)=\int (T_x(z)-m_{\theta }(x){)}^2{P}_{\theta }(dz)\in [0,\mathrm{\infty }]\mathrm{.}$ This is the deterministic-rule specialization of the risk in Loss functions and risk: the action space is $\mathbb{R}$ with its Borel sigma-algebra, the action is $T_x(z)$, and the loss section is measurable because it is continuous in $a$. The integral is nonnegative, so it is defined without a finite-moment assumption.

Martin, §5.1, PDF pp. 100–102; Hoff, §§4.1–4.2, PDF pp. 7–8.