Empirical Risk Equal to True Error under condition [Note: Crosspost from stackoverflow, I think I maybe should've started here]
I've been working through "Understanding Machine Learning: From Theory to Algorithms", and on pg. 46, after the realizability assumption has been dropped from the defintions of loss/risk "L", a note is made that:

where D is an arbitrary joint probability distribution X x Y, X is the feature set, Y is the label set, S is a sample (sampled from the distribution D) of size m, and h is a learned hypothesis.
However, I don't understand this notation - what does it mean to have a probability distribution "uniform over S"? Does this mean that, whatever the distribution D is, that S is sampled uniformly over it according to the probabilities (over the valid domain) defined in D? What happens when the data size m isn't nearly large enough to encapsulate some notion of "uniform"?

As whuber discusses in his comments, this is simply notation to denote that the empirical loss $L_S(h) = \frac{1}{m} \sum_i 1_{h(x_i) \neq f(x_i)}$ is the same as the expected value of the true loss, that is: $L_S(h) = \mathop{\mathbb{E}}_S[L_{D (\text{uniform over S)}}(h)]$.
