True error

Suppose we are in the modelling framework with feature space $A$ with some probability distribution $\mathbb{D}$ on it, a hypothesis space $H$ and the true concept $f: A \rightarrow B$. For some candidate hypothesis $h \in H$ the true error

$$Error_{\mathbb{D}}(h) = \mathbb{P}_{\mathbb{D}}[h(a) \neq f(a)] = \int_{a \in A} \mathbb{I}[h(a) \neq f(a)] d \mathbb{D}.$$

This is the integral of the indicator function on whether it is correct or not weighted with respect to $\mathbb{D}$.