List colorings with measurable sets

The measurable list chromatic number of a graph G is the smallest number ξ such that if each vertex v of G is assigned a set L(v) of measure ξ in a fixed atomless measure space, then there exist sets c(v) ⊆ L(v) such that each c(v) has measure one and c(v) ∩ c(v′) = ∅ for every pair of adjacent vertices v and v′. We provide a simpler proof of a measurable generalization of Hall’s theorem due to Hilton and Johnson [J. Graph Theory 54 (2007), 179–193] and show that the measurable list chromatic number of a finite graph G is equal to its fractional chromatic number.