This tag is for questions relating to the functions of random variables which is a function from $Ω$ into a suitable space of functions (where $Ω$ is the sample space of a probability space that has been specified). Technically, there is also a measurability condition on this function.

A function of an arbitrary argument $~t~$ (defined on the set $~ \mathcal{T}~$ of its values, and taking numerical values or, more generally, values in a vector space) whose values are defined in terms of a certain experiment and may vary with the outcome of this experiment according to a given probability distribution.

In probability theory, attention centres on numerical (that is, scalar) random functions $~X(t)~$; a random vector function $~X(t)~$ can be regarded as the aggregate of the scalar functions $~X_a(t)~$, where $~a~$ ranges over the finite or countable set $~A~$ of components of $~X~$, that is, as a numerical random function on the set $~T_1=T\times A~$ of pairs $~(t,a),~t\in ~T,~a\in~A~.$

For more details, find

$1.~$ "Theory of Random Functions" by A. Blanc-Lapierre & R. Fortet

$2.~$ "The theory of stochastic processes" by I. Gikhman

$3.~$ https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables