Monadic Second-Order Logic with Path-Measure Quantifier is Undecidable

We prove that the theory of Monadic Second-Order logic (MSO) of the infinite binary tree extended with qualitative path-measure quantifier is undecidable. This quantifier says that the set of infinite paths in the tree that satisfies some formula has Lebesgue-measure one. To do this we prove that the emptiness problem of qualitative universal parity tree automata is undecidable. Qualitative means that a run of a tree automaton is accepting if the set of paths in the run that satisfy the acceptance condition has Lebesgue-measure one.