  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.
