nLab essentially bounded function

Definition

Let $X$ be a measure space, or at least a measurable space equipped with a $\delta$-filter of full sets (or a $\sigma$-ideal of null sets). Let $Y$ be a bornological set.

A function from $X$ (or from a full subset of $X$) to $Y$ is essentially bounded if it is almost equal to a bounded function from $X$ to $Y$. That is, for some full subset $E$ of $X$, the direct image $f(E)$ is a bounded subset of $Y$. (Here we equip $X$, and indeed equip every full subset of $X$, with the trivial bornology according to which every set is bounded.)

If $Y$ also has the structure of a measurable space, then we typically require an essentially bounded function to be measurable, although technically this is an independent property. (We can always equip $Y$ with the trivial measurable structure according to which only the empty subset and the improper subset are measurable; then every function to $Y$ is measurable.)

The quotient set of essentially bounded measurable functions from $X$ to $Y$ modulo almost equality is denoted $L^\infty(X,Y)$. With $L^\infty(X)$, we typically take $Y$ to be the real line equipped with the Borel-measurable sets, or occasionally the complex plane. With $L^\infty$, we typically take $X$ to be the real line equipped with the Lebesgue-measurable sets.

Properties

Given any measure space $X$, $L^\infty(X,\mathbb{R})$ and $L^\infty(X,\mathbb{C})$ are (real and complex) Lebesgue spaces with $p = \infty$, hence Banach spaces. In fact, they are Banach algebras, indeed $C^*$-algebras. At least when $X$ is localizable, they are in fact $W^*$-algebras. (Note that the involution $*$ is trivial when the target is $\mathbb{R}$.) Of course, they are commutative; $L^\infty(X,\mathbb{R})$ may also be viewed as an associative JB-algebra (thus a JBW-algebra if $X$ is localizable).

Conversely, every commutative complex $W^*$-algebra is, up to isomorphism of $W^*$-algebras, of the form $L^\infty(X,\mathbb{C})$ for some localizable measure space $X$; the same goes for $L^\infty(X,\mathbb{R})$ and commutative real $W^*$-algebras with trivial involution. In fact, we have more: a dual equivalence between the category of localizable measure spaces (or rather, localizable measurable spaces, as the morphisms —taken up to almost equality, of course— need not respect the measure, except for the full and null sets) and the category of commutative complex $W^*$-algebras (or the category of commutative real $W^*$-algebras with trivial involution, or the category of associative real $JBW$-algebras).

Uselessness

Arguably, we do not need the concept of essentially bounded function; it is sufficient to consider bounded partial functions with full domains, that is almost functions. Traditionally, measure theory uses total functions, but it cares about them only up to almost equality, so almost functions are a natural concept. In constructive analysis, one must use almost functions, since then not every partial function can necessarily be extended to a total function.

Nevertheless, essentially bounded functions are traditional.