Trying to figure out which the training set is
To clarify the proposed classifier (because I think it depends on some nonstandard notation): classify any point that is in the (finite) training set according to its true label.  (Clearly this gives you an error rate of 0 on the training set.)  For any point not in the training set, just predict label 0.
Now, consider a random new point in the gray square: it's the same as a point from the training set with probability 0, so assume it's not.  Then our classifier predicts label 0.  But the point has probability 1/2 (area of blue square over area of gray square) of being label 1.  So the error rate of our classifier, on test data, is 1/2.