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.


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 $L^q$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.