  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.
