A fractional version of the Erdős-Faber-Lovász conjecture

Let H be any hypergraph in which any two edges have at most one vertex in common. We prove that one can assign non-negative real weights to the matchings of H summing to at most |V(H)|, such that for every edge the sum of the weights of the matchings containing it is at least 1. This is a fractional form of the Erdo{combining double acute accent}s-Faber-Lovász conjecture, which in effect asserts that such weights exist and can be chosen 0,1-valued. We also prove a similar fractional version of a conjecture of Larman, and a common generalization of the two.