second order stochastic dominance without the same mean
Let $u(x) = x$ which is increasing and concave. Then the defining condition of SOSD reads

$$\int x\mathrm dF(x)\ge \int x\mathrm dG(x) \implies E_F(X) \geq E_G(X) \tag{1}$$

..which would contradict the case  $E_F(X) &lt; E_G(X)$ that would be permissible under a general "different means" postulate. On the other hand, we see that the "same mean" condition can be violated by the defining condition for SOSD itself. What does that tell us?

1) That $E_F(X) \geq E_G(X)$ is a necessary condition for $F$ to SOSD $G$.  

2) ...And so that the requirement "$F$ and $G$ have the same mean" wrongly restricts the application of the concept of SOSD.