Ambiguity through the lens of measure theory

In this paper, we establish a strong link between the ambiguity for finite words of a B\"uchi automaton and the ambiguity for infinite words of the same automaton. This link is based on measure theory. More precisely, we show that such an automaton is unambiguous, in the sense that no finite word labels two runs with the same starting state and the same ending state if and only if for each state, the set of infinite sequences labelling two runs starting from that state has measure zero. The measure used to define these negligible sets, that is sets of measure zero, can be any measure computed by a weighted automaton which is compatible with the B\"uchi automaton. This latter condition is very natural: the measure must put weight on cylinders [w] where w is the label of some run in the B\"uchi automaton.