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Let $(X, \alpha, \mu)$ be a finite measure space. In the context of a measure theory course, I heard that every function $f \in L_q(\mu)$ can be written as a product of functions $g \in L_p(\mu)$ and $h \in L_s(\mu)$ where $$\frac{1}{q} = \frac{1}{p} + \frac{1}{s}$$ holds. I thought about how to prove that but all I know is how to estimate various inclusions, i.e. $L_q(\mu) \subseteq L_p(\mu)$, for $1\leq p < q \leq \infty $ in the case of $\mu(X) < \infty$. What is the idea here?

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    $\begingroup$ the idea is that $|f|^q=|f|^{\frac{ps}{s+p}}$. Therefore one can define $g=f|f|^{\frac{s}{s+p}-1}$ and $h=|f|^{\frac{p}{s+p}}$. They belong respectively to $L_p$ and $L_s$ and their product is $hg=f|f|^{\frac{s}{s+p} +\frac{p}{s+p}-1}=f$ $\endgroup$ – GGG Feb 7 '17 at 9:20
  • $\begingroup$ Great! If you write it as an answer I can mark it. $\endgroup$ – Taufi Feb 7 '17 at 9:28
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The idea is that $|f|^q=|f|^{\frac{ps}{s+p}}$. Therefore one can define $g=f|f|^{\frac{s}{s+p}−1}$ and $h=|f|^{\frac{p}{s+p}}$. They belong respectively to $L_p$ and $L_s$ and their product is $f$. We can also observe that the hypotesis of $X$ having finite measure was not used.

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  • $\begingroup$ I marked it as accepted answer. Yet, how would you show that $g,h$ belong to $L_p$ and $L_s$ respectively? $\endgroup$ – Taufi Feb 7 '17 at 9:48

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