I am studying for Bartle's book the elements of integration and lebesgue measure. He leaves this passage as an exercise which is to show that there exists $ g \in L_q $ such that it is worth (8.10) which is: $G(f)=\displaystyle \int fg \ d\mu $ with $G$ linear functional. I have tried to define functions of type $g= \{ sgn f \}|f|^{\alpha}$, but I could not get the result. Can someone give a tip?