In this paper, we consider automata accepting irreducible sofic shifts, that
is, strongly connected automata where each state is initial and final. We
provide a characterization of unambiguity for finite words by means of measure
of sets of infinite sequences labelling two runs. 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
