Error rate (modelling)

Suppose we are in the modelling framework and have probability distribution $\mathbb{D}: A \rightarrow [0,1]$ on our domain. We define the error rate to be

$$\mathbb{P}_{\mathbb{B}}[\hat{f}(x) \neq f(x)] = \int_{a \in A} \mathbb{D}(a) \mathbb{I}[\hat{f}(x) \neq f(x)]$$

Where $\mathbb{I}$ is the indicator function. If $A$ is discrete this is simply adding the probability of a value $a \in A$ occurring if our model $\hat{f}$ got it right.