Consistency in Statistical Decision Theory Let $(\mathcal X,\mathcal F,\mathcal P)$ be a statistical model with $\mathcal P = \{P_\theta : \theta\in\Theta\}$. A decision rule is a measurable function $\delta:(\mathcal X, \mathcal F)\rightarrow(\mathcal A,\mathcal G)$. Examples for such decision rules include parameter estimates, or hypothesis tests. This question is concerned with certain properties of decision rules.
Definition. A decision rule is said to be unbiased iff $$\mathbb E_{P_\theta}[L(\vartheta, \delta)]\geq\mathbb E_{P_\theta}[L(\theta, \delta)]$$ for each $\theta,\vartheta\in\Theta$.
Assuming a certain loss function $L:(\mathcal A\times\mathcal A, \mathcal G\otimes\mathcal G)\rightarrow(\mathbb R,\mathcal B)$, this yields the familar concepts for unbiasedness of an estimator and unbiasedness of a hypothesis test:
Theorem 1. Assume the $L_2$ loss $L(t,a) = \Vert t - a\Vert_2$, and $\tau:\Theta\rightarrow\mathcal A$. Then $\delta$ is unbiased iff $\mathbb E_{P_\theta}[\delta] = \tau(\theta)$ for all $\theta\in\Theta$, provided that $\mathbb E_{P_\theta}[\Vert\delta\Vert^2_2] < \infty$, and $\mathbb E_{P_\theta}[\delta]\in\tau(\Theta)$.
Theorem 2. Assume the $0$-$1$-loss $L(t,a) = \ell_0 a\mathsf 1_{\Theta_0}(t) + \ell_1(1-a)\mathsf 1_{\Theta_1}(t)$, where $\ell_0,\ell_1>0$, $\Theta = \Theta_1\cup\Theta_0$ with $\Theta_1\cap\Theta_0=\emptyset$, and $\mathsf 1$ is the usual indicator function. Then $\delta$ is unbiased iff $\mathbb E_{P_\theta}[\delta]\leq\alpha$ for all $\theta\in\Theta_0$, and $\mathbb E_{P_\theta}[\delta]\geq \alpha$ for all $\theta\in\Theta_1$, where $\alpha = \frac{\ell_1}{\ell_0+\ell_1}$.
Another important concept in testing and estimation theory is consistency:

*

*A sequence of estimtors $\delta_n$ is said to be consistent (for $\tau(\theta)$) iff $P_\theta(\Vert\delta_n - \tau(\theta)\Vert\geq\epsilon)$ converges to zero for each $\theta\in\Theta$.

*Similarily, a sequence of tests $\delta_n$ is said to be consistent iff $\sup_{n\in\mathbb N}\mathbb E_{P_\theta}[\delta_n]\leq\alpha$ for all $\theta\in\Theta_0$, and $\mathbb E_{P_\theta}[\delta_n]$ converges to $1$ for all $\theta\in\Theta$.

The concepts are clearly related: for estimators, consistency states that the risk converges to zero in probability. For tests it states that the type I error is bounded, while the probability of a type II error converges to zero. It is thus natural to think about generalizations in the decision theory setup stated above. However, I was not able to do so. Is it even possible?
 A: Definition: We say that a sequence of decisions $\delta_n$ in $\mathcal{A}$ is consistent for the tuple $(L,\tau(\theta),\mathcal{A}',\Theta')$ iff for all $\theta\in\Theta'\subset\Theta$, $\delta_n$ is a sequence in $\mathcal{A}'\subset\{(\mathcal{X},\mathcal{F})→(\mathcal{A},\mathcal{G})\}$ and:
$$
L(\tau(\theta),\delta_n)\rightarrow 0,\quad \text{in $P_\theta$-probability}
$$
Let's check that we can recover estimation and testing with this definition:
Estimation:
Take $L(\tau(\theta),\delta):=||\tau(\theta)-\delta||$, $\Theta' := \Theta$ and $\mathcal{A}':=\mathcal{A}$. Then you recover the given definition of consistency for estimators.
Testing:
Here, we take $L(\tau(\theta),\delta):=1-\delta$, $\Theta' := \Theta_1$
$$
\mathcal{A}':=\{\delta:(\mathcal{X},\mathcal{F})→(\mathcal{A},\mathcal{G}):\sup_{n\in\mathbb{N}}\mathbb{E}_{P_\vartheta}[\delta]\leq \alpha,\forall\vartheta\in\Theta_0\}
$$
What is left to show is that $1-\delta_n\rightarrow 0$ in $P_\theta$-probability is equivalent to $\mathbb{E}_{P_\theta}[\delta_n]\rightarrow1$, for all $\theta\in\Theta_1$.
Using dominated convergence:
$$\begin{align*}
\lim_{n\rightarrow\infty}\mathbb{E}_{P_\theta}[\delta_n]]&=
\lim_{n\rightarrow\infty}\int_0^\infty P_\theta[\delta_n>t]dt\\&=
\int_0^\infty \lim_{n\rightarrow\infty}P_\theta[\delta_n>t]dt\\&=
\int_0^\infty 1\{t<1\}dt=1
\end{align*}$$
