$$E[L] = \sum_k \sum_j \int_{R_j} L_{k,j} p(x, C_k)$$
L is a loss function that returns a real value given a pair (i,j), with i as the index of true class, and j as the index of the predicted class of an input. $p(x, C_k)$ is a joint pdf of a random variables X, which is the input, and C, which is a class. $R_j$ is a decision region j of the input space. My question is how to interpret the right side of the equation as an expectation? L is not a function of x, and so $p(x, C_k)$ is not a corresponding distribution. I expect the relevant pdf should be with respect to the random vector $(C, D)$ with C is the predicted class, and D is the true class of an input.