  Motivated by known results for finite tournaments, we define and study the
score functions of tournament kernels and the degree distributions of
tournament limits. Our main theorem completely characterises those
distributions that appear as the degree distribution of some tournament limit
and those functions that appear as the score function of some tournament
kernel. We also show that only the uniform distribution can be realised as the
outdegree distribution of a unique tournament limit. Finally we define
self-converse tournament limits and kernels and characterise their degree
distributions and score functions.
