Codomain of almost everywhere defined functions

Question

So I am currently wondering about how one can write down the codomain of a function that is only almost everywhere defined. In particular, I am looking at the radon nikodym derivative $f$, which is only defined almost everyhwere. So how could I possibly write down the codomain of this function? Some books write down sth like that it maps to $[0, \infty)$. But there actually might be a nullset where it maps to infinity, right? So then we would have to include that in the codomain, since the codomain should be independent of any measures and of what happens when integrating the function.

Answer 1

Technically, there is no single function which is the Radon-Nikodym derivative. When we say $f$ is a Radon-Nikodym derivative, we mean it's a measurable function $\Omega \to [0, \infty)$ defined everywhere, satisfying the rule $$\nu(A) = \int_A f \ d\mu\tag{1}.$$

The Radon-Nikodym theorem says that if $\nu \ll \mu$, such an $f$ must exist, and any two such $f, g$ must agree $\mu$-almost-everywhere. When we talk about the Radon-Nikodym derivative, we're actually talking about the equivalence class of functions $f$ satisfying $(1)$, not any one specific function.

Edit: to be clear, in general there is no single function which is the derivative, even if that function is allowed to be undefined on a set of null measure. Any two derivatives $f$ and $g$ can disagree at any point $x$ satisfying $\mu(\{x\}) = 0$; if $\mu$ is Lesbegue measure, the two derivatives $f$ and $g$ can disagree anywhere.

When we say the derivative is "defined almost everywhere", it's shorthand for the result above; it's not a literal statement about a specific derivative function.