I thought the statistical definition of entropy for an isolated system of energy E (i.e. microcanonical ensemble) was [itex]S=k \ln \Omega[/itex] where [itex]\Omega[/itex] is the volume in phase space of all the microstates with energy E.

so they use the volume in phase space where energy < E instead of the surface where energy = E. Do these notions coincide? I would think they'd conflict. Why do they say "by the usual definitions", I'm confused.

I'm finding a source (Huang, Statistical Mechanics, 2nd edition, p134) that states [itex]S = k \log \Omega[/itex] and [itex]S = k \log \Sigma[/itex] are indeed equivalent up to a constant dependent of N. The reason for that, I don't seem to get, as the text is a bit too advanced for me atm.

In a way I'm willing to accept the equivalency (as it would clear up my problem), but there's one thing that bothers me: take for example a state of a certain system such that if you lower the energy, entropy goes up (think of a system with bounded energy), doesn't the [itex]\Sigma(E)[/itex] (= the volume in phase space where energy < E) definition make this behavior impossible, because surely (by definition) [itex]E_1 < E_2 \Rightarrow \Sigma(E_1) < \Sigma(E_2) \Rightarrow S(E_1) < S(E_2)[/itex]?