  We generalize to all normal complex algebraic varieties the valuative
characterization of multiplier ideals due to Boucksom-Favre-Jonsson in the
smooth case. To that end, we extend the log discrepancy function to the space
of all real valuations, and prove that it satisfies an adequate properness
property, building upon previous work by Jonsson-Musta\c{t}\u{a}. We next give
an alternative definition of the concept of numerically Cartier divisors
previously introduced by the first three authors, and prove that numerically
Q-Cartier divisors coincide with Q-Cartier divisors for rational singularities.
These ideas naturally lead to the notion of numerically Q-Gorenstein varieties,
for which our valuative characterization of multiplier ideals takes a
particularly simple form.
