# $L^p$-space inclusions

Let $$1\leq p. Which of the following inclusions are true?

1. $$L^p(0,1)\subset L^q(0,1)$$
2. $$L^q(0,1)\subset L^p(0,1)$$
3. $$L^p(0,\infty)\subset L^q(0,\infty)$$
4. $$L^q(0,\infty)\subset L^q(0,\infty)$$

I already know that 1. is false (consider $$f(x)=1/\sqrt{x}$$ with $$p=1$$, $$q=2$$) and 2. holds, which can be shown using the Hölder-inequality.

Now I'm not sure about 3. and 4. I think 3. doesn't hold either, but cannot think of an example to show this. Finally 4. I think is wrong either, since as far as I know the inclusion $$L^q(\Omega)\subset L^p(\Omega)$$ only holds if $$\lambda(\Omega)<\infty$$. But again, I cannot think of a counter-example for this either.

Can anyone think of good examples for this? (Or correct my answer if I'm wrong) Thanks!

• A counterexample for 3. is $f(x) = x^{-1/q} \chi_{(0,1)}$. A counterexample for 4. is $f(x) = x^{-1/p} \chi_{(1,\infty)}$. – Umberto P. Apr 24 '14 at 15:04

Suppose that $(X,\mathcal M,\mu)$ is a measure space and that $1 \le p < q < \infty$. Then
1. $L^q(\mu) \subset L^p(\mu)$ if and only if $X$ does not contain sets of arbitrarily large finite measure, and
2. $L^p(\mu) \subset L^q(\mu)$ if and only if $X$ does not contain sets of arbitrarily small positive measure.