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I am stuck on this.

Let $(X, \mathcal{F}, \mu)$ be a measure space satisfying $\mu(X)<\infty$ and let $f, g, h: X \rightarrow \mathbb{R}$ be measurable functions such that $f^{4}, g^{4}$ and $h^{3}$ are Lebesgue integrable. Must $f g h$ be Lebesgue integrable? Justify your answer.

I had the idea to prove that the absolute value of fgh is integrable using the Holder integral (I separated fg from h by using p=4 and q=4/3 but I got stuck with the integrale of a power of h).

Any hint or explanation would be appreciated.

ReDK121
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Since $\mu (X)<\infty$ we have inclusion $L^p (\mu )\subset L^s (\mu )$ for $p>s. $So we have $f,g\in L^4 (\mu )$ hence $f,g\in L^3 (\mu )$ and thus $f,g,h\in L^3 (\mu ) $ but this implies that $fgh\in L^1 (\mu )$ thus the function $fgh$ is integrable.

  • Thank you so much. I don't know how that slipped by me (the proof of it has the Holder inequality too that I was trying to use in such a complicated way). – ReDK121 Nov 08 '21 at 15:50