Random variables: functions or equivalence classes

Question

Suppose, we have a continuous parameter stochastic process. Should I consider each of the random variables as functions or equivalence class (in a.e. sense) ?

Answer 1

The vast majority of results about probability theory are formulated in a sense such that the variables considered are actual measurable functions and not a.e. equivalence classes, even though many results of interest about these variables can be lifted to results about a.e. equivalence classes.

Continuous-parameter stochastic processes are no exception. Thus a Brownian motion $(B_t)_{t\ge0}$ is not a collection of equivalence classes, it is a collection of actual mappings $B_t: \Omega \mapsto \mathbb{R}$ from some background probability triple $(\Omega,\mathcal{F},P)$ to the real numbers.

This actually also serves a particular purpose. If each $B_t$ was an equivalence class, it would complicate our ability to speak of path properties: Speaking of "continuity" of a collection of a.e. equivalence classes is inconvenient. However, having each $B_t$ be a function from $\Omega$ to $\mathbb{R}$, the statement that $t\mapsto B_t(\omega)$ is continuous for all $\omega\in\Omega$ or almost all $\omega\in\Omega$ is entirely well-formed.

So, summing up, there's very rarely any particular reason for you to think of stochastic variables - alone or indexed by a continuous parameter - as equivalence classes instead of actual mappings.