As usual denote $L^p$ the quotient space where two integrable functions are identified if they are equal almost everywhere. So I'm using the definition written here:
http://en.wikipedia.org/wiki/Lp_space
Then we have the following result: for each $p \geq 1$ we have $L^{\infty}(X) \subseteq L^{p}(X)$ where X is a finite measure space and $L^{\infty}$ denotes the set of all essentially bounded functions endowed with the $||f||_{\infty}$ pseudonorm.
So I took $f \in L^{\infty}(X)$ then by definition there is some bounded function $g$ such that $g=f$ a.e. But then $f=g$. So:
$\int |f|^{p} = \int |g|^{p} \leq \int (||g||_{\sup})^{p} < \infty$.
Questions: Is the above correct? Why do we need $p \geq 1$. Why wouldn't $p>0$ work? Is it because we need $p \geq 1$ in the case q is not $\infty$ or where exactly?
Thank you.