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.(adsbygoogle = window.adsbygoogle || []).push({});

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 thegdetermined 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!

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# A question on proof of Riesz Representation Theorem when p=1

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