Notation used for multivariate random variables Let $(\Omega, \mathcal{F}, P)$ be a probability space and $X_1(\omega), \dots, X_n(\omega)$ be random variables defined on the space. Suppose we are concerned with the joint behavior of the variables. Then one usually considers the variables as a single multivariate random variable defined as a vector of (one-dimensional) random variables: $\mathbf{X}(\omega) = (X_1(\omega), \dots, X_n(\omega))^T$.
My first set of questions is: Is it necessary to pack random variables into a random vector to have a legitimate (mathematically correct) discussion about their joint distribution? Can one consider them as just a set $\{ X_1(\omega), \dots, X_n(\omega) \}$?
A vector induces a particular order on its elements while a set does not. It is quite common to denote, for example, the joint PDF as $f_{X_1, \dots, X_n}(x_1, \dots, x_n)$ where, again, a particular order is implied even though the vector notation is not used explicitly. So, is it all about ordering?
My second set of questions is: Is it correct to denote a set of random variables as $\text{(a-regular-weight-letter-goes-here)}(\omega)$, for example, $X(\omega) = \{ X_1(\omega), \dots, X_2(\omega) \}$? (The question is mainly concerned with the use of $\omega$ next to a set of random variables.) Then is it acceptable to denote, for instance, the corresponding PDF by $f_X(x)$ meaning that $x = \{ x_1, \dots, x_n \}$? Since the order is not specified, it is presumably wrong.
Thank you.
Regards,
Ivan
 A: To be concrete, assume every $X_i$ is a real valued random variable. Then a first observation is that you confuse $X_1(\omega)$, say, which is a number, and $X_1$, which is a function. For example, the random vector you consider is $X=(X_1,\ldots,X_n)$ (with or without transpose), which is a function from $\Omega$ to $\mathbb R^n$, certainly not $X(\omega)=(X_1(\omega),\ldots,X_n(\omega))$, which, for each $\omega$ in $\Omega$, is a point in $\mathbb R^n$.
Now to the main part of your question. Consider some real numbers $x_k$. There is definitely a difference between $x=(x_1,\ldots,x_n)$, which is a point in $\mathbb R^n$, and $u=\{x_1,\ldots,x_n\}$, which is a subset (of size at most $n$) of $\mathbb R$. The former determines the latter but the latter carries strictly less information than the former.
For example, several points $(x_1,x_2,x_3)$ in $\mathbb R^3$ correspond to $u=\{3,17,42\}$ and several points $(x_1,x_2,x_3)$ in $\mathbb R^3$ correspond to $u=\{17,42\}$ (but only one point corresponds to $u=\{42\}$). The same applies to random variables.
To sum up, the object quasi universally considered in such a setting is the vector $X=(X_1,\ldots,X_n)$ (and its distribution on $\mathbb R^n$, say). One can translate these, with some loss of information, in terms of the random set $U=\{X_1,\ldots,X_n\}$ (and in terms of its distribution on the space of nonempty finite subsets of $\mathbb R$, endowed with its natural sigma-algebra) but this is almost never done a priori.
