Definition of $L^p$ as a set Looking at Functional Analysis, I know that the space of convergent sequences $\ell^p$ is defined to be $$\ell^p := \left\{\left(x_k\right)_{k=1}^\infty\mid\sum_{k=1}^\infty |x_k|^p<+\infty\right\}.$$ My question is what is the set definition of the space of functions $L^p$ ? I'm not looking for the most general definition, just something to get me started. Strangely enough I can't find such an explicit definition on the net. I'm guessing it will be something like: $$L^p:=\left\{f\in \text{C}([0,1],\mathbb{C})\mid\|f\|_p^p<+\infty\right\},$$ for some norm $\|\cdot\|_p$, i.e. the set of continuous function $f:[0,1]\to\mathbb{C}$ whose chosen norm is finite. I think this norm is $$||f||_p^p=\int_{\mathbb{R}}|f(x)|^pdx,$$ by analogy with the square integrable functions. Just wanted to check if this was correct or not?
 A: The functions belonging to $L^p$ are in general not continuous. And, to be precise, the elements of $L^p$ are not even functions, but equivalence classes of functions.
First we start with the space $\mathscr{M}$ of measurable functions. Measurable functions can be pretty wild, but they have enough regularity that one can handle them.
For a measurable function $f$, the function $\lvert f\rvert^p \colon x \mapsto \lvert f(x)\rvert^p$ is also measurable, and non-negative. Hence the integral
$$\int_\mathbb{R} \lvert f(x)\rvert^p\,dx$$
is well-defined as an element of $[0,\infty]$. Then we define the space
$$\mathscr{L}^p = \left\{ f\in \mathscr{M} : \int_\mathbb{R} \lvert f(x)\rvert^p\,dx < \infty\right\}$$
of $p$-integrable functions. On $\mathscr{L}^p$, we have a seminorm
$$\lVert f\rVert_p := \left(\int_\mathbb{R} \lvert f(x)\rvert^p\,dx\right)^{1/p}.$$
But, that seminorm does not distinguish between functions that differ only on a set of measure $0$, and thus the topology induced by the seminorm $\lVert\,\cdot\,\rVert_p$ on $\mathscr{L}^p$ is not a Hausdorff topology, which makes dealing with these spaces somewhat awkward. To remedy that, we consider the space of equivalence classes of $p$-integrable functions modulo being equal "almost everywhere". In terms of functional analysis/linear algebra, we consider the quotient
$$L^p = \mathscr{L}^p/\mathscr{N},$$
where
$$\mathscr{N} = \left\{ f\in\mathscr{M} : \int_{\mathbb{R}} \lvert f(x)\rvert\, dx = 0\right\}$$
is the space of negligible functions (functions that are $0$ almost everywhere), to obtain a (complete) normed space [for $p \in [1,\infty)$, the spaces $\mathscr{L}^\infty$ and $L^\infty$ are not defined by a finite integral but by being "essentially bounded"], which is Hausdorff, and with which one can deal much better.
These constructions generalise to other measure spaces, without any difference in principle. The spaces $\ell^p$ are the special case where the domain is $\mathbb{N}$ and the measure is the counting measure. Since the counting measure has no null sets except the empty set, $\mathscr{N} = \{0\}$ for these.
