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Let $(X,\mathcal{M},\mu)$ a positive measure space, then $L^r(\mu)\subset L^p(\mu)+L^q(\mu)$. How can I prove this?

I know the standard inclusions for finite measure spaces, and spaces without subests of arbitrary small measure. But I don't know what kind of inequality I can use in this case.

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I assume you mean that $p < r < q$.


The higher $p$ is, the better job $L^p$ does of detecting large values, and the smaller $p$ is, the better job $L^p$ does detecting large support.

So given $f \in L^r$, consider cutoffs:

$$f = f \chi_{|f| > 1} + f \chi_{|f| \le 1}$$

The first has small support, and is easily seen to be in $L^p$. The second has small values, and is easily seen to be in $L^q$.

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