  We study three models of the problem of adversarial training in multiclass
classification designed to construct robust classifiers against adversarial
perturbations of data in the agnostic-classifier setting. We prove the
existence of Borel measurable robust classifiers in each model and provide a
unified perspective of the adversarial training problem, expanding the
connections with optimal transport initiated by the authors in previous work
and developing new connections between adversarial training in the multiclass
setting and total variation regularization. As a corollary of our results, we
prove the existence of Borel measurable solutions to the agnostic adversarial
training problem in the binary classification setting, a result that improves
results in the literature of adversarial training, where robust classifiers
were only known to exist within the enlarged universal $\sigma$-algebra of the
feature space.
