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Could someone confirm or refute the following statement?

[tex]f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right)[/tex]

where [tex]1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1[/tex] and [tex](X, \mu)[/tex] is a measurable space (of course, the [tex](\Rightarrow)[/tex] is trivial by Holder inequality)

Thanks in advance!

[tex]f \in L^p\left(X, \mu\right) \: \Leftrightarrow \: \int_X{\lvert fg \rvert d\mu < \infty\: \forall g \in L^q\left(X, \mu\right)[/tex]

where [tex]1<p<\infty,\: \frac{1}{p}+\frac{1}{q}=1[/tex] and [tex](X, \mu)[/tex] is a measurable space (of course, the [tex](\Rightarrow)[/tex] is trivial by Holder inequality)

Thanks in advance!

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