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. [URL]http://i3.6.cn/cvbnm/ac/9a/a3/3d06837bc78f74ba103b6d242a78e3a1.png[/URL] The equation (8.10) is [tex]G(f)=\int fgd\mu[/tex]. The definition of [tex]L^\infty[/tex] space is as follows: [URL]http://i3.6.cn/cvbnm/a5/0e/22/259193b92d8d2ef4878532eefec4d900.png[/URL] 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!