$ \newcommand{\MM}{\mathcal{M}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\set}[1]{\left\{ #1 \right\}} $In my analysis class, we have been discussing certain properties of the set of measurable functions; to introduce some notation, we'll define $$ \mathcal{M} := \MM(E) := \left\{ f : E \subseteq \R^n \to \R \; \middle| \; f \text{ is measurable} \right\} $$ under the usual definitions of measurability. We have observed the following properties in this class:
- $f,g \in \MM \implies f+g \in \MM$
- $f \in \MM, \lambda \in \R \implies \lambda f \in \MM$ and $f+\lambda \in \MM$ (where $f+\lambda := \{f(x)+\lambda\mid x \in E\}$
These in particular ensure that $\MM$ is a vector space. But we also have
$f,g \in \MM \implies f\cdot g \in \MM$
$f,g \in \MM$, and $g \ne 0$ a.e. $\implies f/g \in \MM$. (At least, provided we restrict the domain of the quotient to avoid $g(x) = 0$, so it might be a different domain than $f,g$ have. On the other hand, of course, if $g(x) \ne 0$ everywhere, then $f,g \in \MM \implies f/g \in \MM$ very literally. So not quite an invertible multiplication.)
$\displaystyle \set{f_k}_{k\in\N} \subseteq \MM \implies \sup_{k \in \N} f_k \in \MM$, and $\displaystyle\inf_{k \in \N} f_k \in \MM$, and $\displaystyle\limsup_{k \to \infty} f_k \in \MM$, and $\displaystyle\liminf_{k \to \infty} f_k \in \MM$
If the limit exists a.e., in particular, then the above gives $\displaystyle \set{f_k}_{k\in\N} \subseteq \MM \implies \lim_{k \to \infty} f_k \in \MM$
This seems like a particularly rich structure, moreso than your average vector space -- we also have closure under multiplication (and it presumably has an inverse). We also have that suprema and infima of sequences, and likewise the limit suprema and infima, also are in the set.
Of course, we can also introduce the $L^p$ norms and make this into whichever we need among a normed space, inner product space, and metric space.
What is the name for such a structure (if one exists), one satisfying conditions such as these? (Perhaps not all of them necessarily, but more than just your usual vector space. Perhaps mentioning other facts or operations as necessary that also hold for $\MM$.)
What would be some other examples of such structures? (Examples other than those tied to measurable functions and measurability, if possible.)
Update: (April 2nd, 2021)
One term that could apply, as noted in the comments, is the notion of an algebra over a field, and a commutative one in particular. We could even say it's unital ($f \equiv 1$ is the obvious identity).
Once we equip the vector space $\MM$ with the usual pointwise multiplication, we have this structure, but there's a lot more to this, namely with the convergence properties.
To some degree, as I've mulled over this, I wonder if they somehow could reflect a "closedness" or "sequential compactness" property. Of course, these are topological properties, and thus we would need to induce a topology. The continuous functions on a compact set have one induced by the supremum norm. Taking a compact domain, then, perhaps $\MM$ would admit these as a subalgebra, since continuous functions are measurable, and that would lead somewhere?