This question comes from the proof of Riesz Representation Theorem in Bartle's "The Elements of Integration and Lebesgue Measure", page 90-91, as the image below shows. The equation (8.10) is [tex]G(f)=\int fgd\mu[/tex]. The definition of [tex]L^\infty[/tex] space is as follows: My question is: why the g determined by Radon-Nikodym Theorem is in [tex]L^\infty[/tex]? I can only prove that it is Lebesgue integrable, that is, belongs to [tex]L^1[/tex] space, but the proof mentions no word on why it is in [tex]L^\infty[/tex], that is, bounded a.e.. Could you please tell me how to prove this? Thanks!