Why is the H theorem is called the big eta theorem?

In France, they refer to the H-theorem of Boltzmann (Théorème H) as 'eta'-theorem (théorème 'eta'). The connection obviously comes from the uppercase version of the Greek letter $$\eta$$, which looks like a capital H. Why would they associate H to $$\eta$$ (?) and not to $$H$$ as Hamiltonian?