Meaning of ‘almost-everywhere constant’ random variable

measure-theoryprobability theoryrandom variables

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

  1. Is the above interpretation correct?

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

  3. 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 $\tilde x : \Omega \rightarrow \{c\}$.

Best Answer

  1. 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).

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

  3. 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..

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