Is the p-value a good measure of evidence? An asymptotic consistency   criterion
  What are the criteria that a measure of statistical evidence should satisfy? It is argued that a measure of evidence should be consistent. Consistency is an asymptotic criterion: the probability that if a measure of evidence in data strongly testifies against a hypothesis H, then H is indeed not true, should go to one, as more and more data appear. The p-value is not consistent, while the ratio of likelihoods is. 
M. Grendar
math.ST stat.TH