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An old assignment question I haven't been able to fully crack goes as follows:

Let $p \in [1,\infty]$ and $q$ its conjugate exponent. Suppose that $g$ is a measurable function such that $\forall f \in L^q$, $fg \in L^1$. Show that there exists an $h \in L^p$ such that: $$ \int f(g-h) =0 \qquad \forall f \in L^q$$ Thus far, I figured I might be able to do something by defining the linear function $S: L^q \to L^1$ by $f \mapsto gf$, which can be easily shown to be continuous by the Closed Graph Theorem. As mentioned in the answers below, the next step is an application of the Riesz Representation theorem, provided I can show that: $$ \int T(g) = \int fg $$ Is bounded. For anybody who is wondering, the details are present here: If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$?

Thanks.

rednexela1941
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  • Seems like the map $f\mapsto \int gf$ is a continuous linear functional on $L^q$, so the Riesz representation theorem implies it is equal to the map $f\mapsto \int hf$ for some $h\in L^p$. – user254433 Feb 23 '18 at 06:51

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By your assumption $L:f\mapsto L(f):=\int gf$ is a linear functional on $L^q$. Now if you can show $L$ is bounded, then Riesz Representation Theorem gives you the existence of $h\in L^p$ such that $L(f)=\int hf$ for every $f\in L^q$.

Rgkpdx
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