Alternatives to "measures":

• Density functions: Most likely false.

• Measures: Perhaps only for rational h?

• Distributions of bounded order: Better chance?

• Integrals against characters: This is the classical Reshetikhin-Turaev theory, the tetrahedron is the quantum 6j symbols!

• Infinite order operators near e (a.k.a. U(g)): This is Drinfel'd's theory of associators!

• "Diagrammatic U(g)": This is the Kontsevich Integral!