$L^1$ space

Definition

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a measurable set and $\lambda$ the Lebesgue measure on ${\mathbb{R}}^n$. The $L^1$ space is:

where $f\sim g$ if $f=g$ almost everywhere (i.e., they differ only on a set of measure zero). Thus, elements of $L^1(\mathrm{\Omega })$ are equivalence classes of functions; the term Lebesgue integrable applies to a representative function $f$, not to the class itself.

The associated norm on the equivalence class $[f]$ is:

which is well-defined because modifying $f$ on a set of measure zero does not change the integral.

The most crucial part of the definition is the quotient by the equivalence relation $\sim$. If you only considered the set of functions with finite integrals without taking the equivalence classes, the mapping $\parallel f{\parallel }_1={\int }_{\mathrm{\Omega }}|f|d\lambda$ would only be a seminorm, not a true norm. This is because a function could be non-zero on a set of measure zero, yet its integral would still be $0$, violating the norm property that $\parallel f\parallel =0⟺f=0$.

Key Properties

• Because of the identification of almost everywhere equal functions, $L^1(\mathrm{\Omega })$ consists of equivalence classes. Consequently, pointwise values like $[f](x)$ are not well-defined for a given $x\in \mathrm{\Omega }$; changing the value of a representative on a set of measure zero yields a different representative but the same equivalence class. And taking a classical derivative is not meaningful in $L^1$. This can be fixed with the embedding of $L^1$ into the space of distribution (functional analysis).
• $L^1(\mathrm{\Omega })$ is a Banach space.