7

I apologize for the terribly worded title, but I didn't know how else to title this questions (which comes from Rudin's Real & Complex Analysis chapter 3 questions).

The question says:

For some measures, the relation $r<s$ implies $L^r(\mu)\subset L^s(\mu)$; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\neq s$. Give examples of these situations, and find conditions on $\mu$ under which these situations will occur.

I am having an extremely difficult time grasping any/everything having to do with measures and Lp spaces, so this question is mind-crippling. I don't even know where/how to begin thinking about such a question. Any input/help/criticism is greatly appreciated.

PiTheMathemagician
  • 571

1 Answers1

4

1) By Holder's Inequality it is immediate to show that $L^p(\Omega) \subset L^q(\Omega)$ for $p \ge q$ if you consider the usual Lebesgue measure on a set of finite measure.

2) Note that $\sum x^p$ converges if $\sum x^q$ converges provided $p \ge q$. This means that $\ell^q(\mathbb{N}) \subset \ell^p(\mathbb{N})$, which we can rewrite as $L^q(\mathbb{N},\#) \subset L^p(\mathbb{N},\#)$, where $\#$ is the counting measure.

3) Consider the Lebssgue measure on an set of infinite measure. This is an example where there's no inclusion of the $L^p$ spaces, but it is trickier to prove.

I hope this helps! :)