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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.

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    @user:Technically, $g=f$ a.e. means that *the equivalence classes* of $f$ and $g$ are equal, rather than $f$ and $g$ necessarily being equal. We usually abuse notation in $L^p$ spaces and use the function to denote its class, but here it might be somewhat confusing to say "$f=g$ a.e., so $f=g$".2010-11-10

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