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

Ambiguity through the lens of measure theory