Weak learner

Suppose we are in the modelling framework a model $\hat{f}$ is called a weak learner if for all probability distributions $\mathbb{D}: A \rightarrow [0,1]$ there exists a suffciently small $\epsilon$ such that

$$\mathbb{P}_{\mathbb{D}}[\hat{f}(x) \neq f(x)] \leq 0.5 - \epsilon.$$

That is its error rate is just bellow chance.