Conditional misclassification risk ($L(C)=\mathbb{P}(C(X)\ne Y\mid \mathcal{T})$)

For a random classifier $C$ determined by training information $\mathcal{T}$, and an independent test pair $(X,Y)$, its conditional misclassification risk is $L(C)=\mathbb{P}(C(X)\ne Y\mid \mathcal{T})$. It is itself a training-dependent random variable. The tower property of conditional expectation gives $\mathbb{E}L(C)=\mathbb{P}(C(X)\ne Y)$.