We prove that the associate space of a generalized Orlicz space L^ϕ(·) is given by the conjugate modular ϕ* even without the assumption that simple functions belong to the space. Second, we show that every weakly doubling Φ-function is equivalent to a doubling Φ-function. As a consequence, we conclude that L^ϕ(·) is uniformly convex if ϕ and ϕ* are weakly doubling.