\begin{definition}[Definition:Lebesgue Space]
{{wtd|Rigorous linking, maybe splitting, but it is vital to introduce $\LL$ and $L$ both on this page}}
Let $\struct {X, \Sigma, \mu}$ be a measure space, and let $p \in \R$, $p \ge 1$.
The '''(real) Lebesgue $p$-space of $\mu$''' is defined as:
:$\map {\LL^p} \mu := \set {f: X \to \R: f \in \map \MM \Sigma, \ds \int \size f^p \rd \mu < \infty}$
where $\map \MM \Sigma$ denotes the space of $\Sigma$-measurable functions.
On $\map {\LL^p} \mu$, we can introduce the $p$-seminorm $\norm {\, \cdot \,}_p$ by:
:$\forall f \in \LL^p: \norm f_p := \paren {\ds \int \size f^p \rd \mu}^{1 / p}$
Next, define the equivalence $\sim$ by:
:$f \sim g \iff \norm {f - g}_p = 0$
The resulting quotient space:
:$\map {L^p} \mu := \map {\LL^p} \mu / \sim$
is also called '''(real) Lebesgue $p$-space'''.
\end{definition}