Meaning of 'almost-everywhere constant' random variable My question stems from page 2 of this paper by Bucy, which states:

[A random variable $x$] is almost everywhere constant a.e.
$P$.

where $P$ is a probability measure. My interpretation of this is as follows (where I consider $x$ to be real-valued).
Interpretation
Given the probability space $(\Omega ,\mathcal F, P)$ and some constant $c\in \mathbb R$, $x:\Omega \rightarrow \mathbb R$ is a random variable of the following form:
\begin{align}
x(\omega) = \begin{cases}
c \qquad&\omega \in A \\
g(\omega) &\omega \in A^c
\end{cases} 
\end{align}
where $A\in \mathcal F$ is such that $P(A^c)=0$ and $g$ is an arbitrary real-valued function. The sets $A$ satisfying the foregoing condition capture what we mean by "almost everywhere" w.r.t. the measure P.
Questions

*

*Is the above interpretation correct?


*Is this to be thought of as a general form of what one might call a 'degenerate' random variable?


*If (2) is yes, then is there some intuition for why such a definition might be desirable? As opposed to defining a degenerate r.v. as the constant function $\tilde x : \Omega \rightarrow \{c\}$.
 A: *

*It means that there exists $c \in \mathbb{R}$ such that $X = c$ a.s.. It's essentially what you wrote, except $g$ is measurable (if $g$ is allowed to be arbitrary, $X$ might not even be measurable).


*Yes I think this is the meaning of the term "degenerate".


*People almost always use equivalence classes for random variables for many reasons. One reason is that this makes $L^p$ spaces normed spaces. Another reason is that most of the time you are working with the measure $P$ and the best you can do is prove that something is true $P$-a.s., e.g. a sequence converges a.s..
A: *

*The interpretation is correct, although I would add for completeness the function $g$ should be measurable, though this is often understood even without specifying it. A more succinct way to say the same thing is that $X$ is constant $P$-a.e. if for every $c\in \mathbb R$, $P(X = c)\in\{0,1\}$.


*It is certainly not a "random" r.v. in the colloquial sense of the word. A common word that is used to describe such r.v.'s is deterministic random variables.


*In measure theory and probability theory, it is quite common to consider measurable functions (or random variables) as only being defined up to sets of measure zero, and to work instead with equivalence classes of (measurable) functions that are equal to one another almost everywhere. I think in practice, it's harmless to assume that a random variable which is constant almost everywhere is in fact constant everywhere, though I can't imagine a situation where it makes anything clearer. Indeed, the sample space (domain) usually plays a minor role relative to the distribution or law of a random variable we are considering (i.e., the numbers $P(X\in A)$, as $A$ ranges over the measurable sets in the codomain of the r.v.).
