With s>1 and r>s. How can I show that $\displaystyle L^s \supset L^r$, for expectation with respect to a probability measure?
Hello,
Show what you've done. Basically, you just have to apply Holder's inequality to $\displaystyle E[X^s]$, with $\displaystyle X^r$ as f and 1 as g. If X is in $\displaystyle L^r$ then $\displaystyle E[X^r]<\infty$, that's how it'll work.
Give it a try.
This is exactly it !
Now you have to talk a little :
Let's take $\displaystyle Y\in L^r$. Then $\displaystyle E[|Y|^r]<\infty$. - your calculations - . Thus $\displaystyle Y\in L^s$, which implies that $\displaystyle L^r\subset L^s$ : any element of the first belongs to the second.