What is space?

For $1 \leq p < \infty$, $\mathcal L^p(\mu)$ is the set of measurable $f$ with $\norm f_p = \bigl(\int\abs f^p\dd\mu\bigr)^{1/p} < \infty$, and $\mathcal L^\infty(\mu)$ the set of $f$ bounded outside a null set, with $\norm f_\infty$ the essential sup — the least $M$ with $\abs f \leq M$ a.e. (the inf is attained: intersect the null sets for $M + \frac1n$). Since $\norm f_p = 0$ only forces $f = 0$ a.e. (Exercise 10.5), we define

elements are classes of functions modulo null sets, and $\norm\cdot_p$ is a genuine norm on $L^p$.