In the definition of an $L^p$ space, do we assume $p$ is an integer, or at least rational?

Question

Are $L^p$ spaces defined only when $p$ is an integer? Are rational numbers also acceptable powers? The definition in Rudin's Real and Complex Analysis doesn't seem to specify limitations on $p$ except that $0<p<\infty$, but I am confused about what it would mean if $p$ were irrational.

Answer 1

$L^p$ spaces make sense as long as $p$ is a positive real - nowhere in the definition is anything more than this required.