Doubt in understanding definition of random variable

Question

I have recently started studying probability. Kindly explain me following

Random variable means a function from $\Omega \to R$(set of real numbers). This is what i understood from my school books. But when i started reading Ross, it is given as in addition to above, any Borel set in R must have preimage in $\Omega$ i.e, preimage of Borel set in R must be an event.

Why this 2nd part is needed? Does it mean Random function(variable) X is "onto"

Another doubt, I thought random variable is some variable. But i understood now that it is infact a function. Then why it is called random "variable"?

If X, Y are 2 random variables, i know that XY is also random variable

My proof:

$X: \Omega \to R \\ Y: \Omega \to R $

then

$XY(\alpha) = X(Y(\alpha)), \alpha$ is some event in $\Omega$ sample space. Since composition of functions is a function from same domain $\Omega$ to R and so XY is a function and so it is a random variable. Please correct me if i am wrong.

Answer 1

There is a common saying: "A random variable is neither random nor a variable." The terminology might have come from some intuition at a non-measure-theoretic level, but in the formal definition it is simply a [measurable] function.

This "measurability" (preimage of Borel set must be measurable in $\Omega$) is important because you would like to assign a probability to events of the form $\{X \in B\}$ for Borel sets $B$. This probability is assigned from a probability measure on $\Omega$. If the preimage $X^{-1}(B)$ is not measurable in $\Omega$, then you cannot assign a probability to that event.

$XY$ is defined as the product $X(\alpha) Y(\alpha)$, not the composition.

Answer 2

The measure-theoretic restriction is to avoid pathological situations where $X \ge c$ can't be assigned a probability because the set $\{\alpha: X(\alpha) \ge c\}$ is not measurable.

Although in principle the random variable $X$ is a function on the sample space $\Omega$, in practice probabilists rarely think of it that way, in fact they often don't bother spelling out what the sample space is.

Your proof makes no sense because $X Y$ is not a composition, it is just the ordinary product of real functions: $(XY)(\alpha) = X(\alpha) Y(\alpha)$.