Notation for the set of measurable functions and the related quotient space

Question

Let $X$ and $Y$ be measurable spaces. Two very quick questions:

1. How do you denote the set of measurable functions from $X$ to $Y$? I usually write $\mathrm{meas}(X,Y)$, but I’d like to be presented with nicer alternatives.

2. Let $\sim$ be the almost-everywhere-equality equivalence relation. How do you denote the quotient space ${\mathrm{meas}(X,Y)}/{\sim}$? I’m trying to get rid of that tilde and possibly find a notation which is “self-explanatory”, but I find things like ${\mathrm{meas}(X,Y)}\big{/}{\,\stackrel{\mathrm{a.e.}}{=}}$ very ugly.

Answer 1

Ok, I'll write it as an answer. When I took a course in measure theory we were denoting the measurable functions by $L^0(X,Y)$, this is like the basic $L^p$ space. So you might try using it.

As for the second question, it was mentioned in the comments that some are using the notations $\mathcal{L^p}$ (including $\mathcal{L^0}$) for the functions and $L^p$ for equivalence classes. This is indeed an option. I don't really use it though, I use the notation $L^p$ for both of them. In many cases it is just not that important if the objects are functions or equivalence classes, and when it's important I just write "here I think of the elements of $L^p$ specifically as equivalence classes" or the other way around.

Answer 2

I second Mark's answer.

In addition, here is a humble suggestion: in analogy with how in topology $hTop(X,Y)$ is commonly used to denote the space of maps up to homotopy (= homsets in the homotopy category) one may use $aeL^0(X,Y)$ if there is a need to distinguish between functions and functions identified a.e.. One drawback of this notation is when there are multiple measures (or measure classes) involved, in which case probably $L^0(X,Y)/\mu$ is more convenient (and more in line with quotienting in the algebraic sense).

In practice I use measures syntactically as modifiers of the standard set-theoretical notations, e.g. "$E=_\mu X$" means "$E$ is full-$\mu$-measure" or "$\forall x\in_{\text{a.e.}} X$" means "for almost all $x$ in $X$" (in case I don't want to name the measure).

Also, to my mind, the litagure æ (with TeX code \ae) is more tasteful and economic than a.e. (though in handwriting it looks like an $x$ or a $\kappa$ which might cause confusion when communicating).