The standard definition is the one given in the two bullet points at the top of the article we are discussing. Or, in more detail, at https://en.wikipedia.org/wiki/Random_variable#Definition.
> The random variable X is a function mapping the outcome to its probability.
you've stated: "The standard definition is the one given in the two bullet points at the top of the article we are discussing" ie:
> The random variable X itself, that is, the function which maps sample space outcomes to numbers.
you've also stated: "Or, in more detail, at (wikipedia link)" which has:
> A random variable is a measurable function from a sample space as a set of possible outcomes to a measurable space
These all appear to be in rough alignment .. all three agree upon a function mapping from outcomes to measure.
You've described the first as "This is simply wrong. Random variables are not defined this way." Perhaps you can expand on why this is so wrong compared to the other two definitions.
> the function which maps sample space outcomes to numbers,
the "numbers" referred to in the codomain are not probabilities. Similarly, in the Wikipedia definition,
> a measurable function from a sample space... to a measurable space,
the "measurable space" codomain has nothing to do with probability of events. Random variables are related to probability distributions/measures, but they are not the same kind of mathematical object.
As a concrete example, consider gambling on a fair coin flip. We can bet 1 unit of money on Heads and define a random variable for the amount of money we win/lose. First we set up the probability space. The sample space is the 2-element set of coin sides, Ω = {H, T} (Heads and Tails). The event set (aka sigma-algebra) F is the power set of Ω: F = {{}, {H}, {T}, {H, T}}. The probability measure assigns probability 0 to the event {}, probability 0.5 to each of the events {H}, {T}, and probability 1 to the event {H, T}.
Now, we can define the random variable of our winnings by the function X mapping Ω to the real numbers by Χ(ω) = 1 if ω = H, -1 if ω = T.
So, within the setting of the probability space (Ω, F, P), the random variable is defined by the function X, which does not map anything to a probability. It maps coin sides to real numbers, including negative ones!
We have not yet constructed any object that gives the probabilities of the possible values of X. The only thing that "maps an outcome to a probability" in our setup so far is the measure P, but that is defined independently of our numeric random variable. If we want to talk about the probabilities that X takes the values -1 and 1, we are talking about the pushforward measure of P by X. This is another mathematical object, distinct from the random variable X itself.
This is one of the points of the article. Suppose we define another random variable Y for the winnings of the person we are gambling against: Y(ω) = -1 if ω = H, 1 if ω = T. Now X and Y are two different random variables, but they have the same distributions! The pushforward measures of P by X and Y both put probability 0.5 on the measurable sets {-1} and {1}.
Is that a massive assumption that such a thing exists that makes the GP comment egregiously wrong, or more of an impatient jump to the end as such a thing always (?) exists?
Do they?
What about? “The random variable X is a function mapping each outcome to the age of one of my cousins”?
Does that seem like a valid definition of the concept of random variable to you because the age of my cousins are numbers - just like probabilities mentioned by glitchc’s comment are numbers?
"probabilities" and "age of one of my cousins" do not sound at all similiar.
One has the feel of a potentially continuous interval of values that can be mapped | scaled to perhaps match up with other forms of expression, the other is a finite discrete set of probably integers between zero and 120 with total set size likely less than a 100.
The point is that the random variable is about mappings to general numbers.
“The random variable X is a function mapping the outcome to its probability”
makes as much sense as
“The random variable X is a function mapping the outcome to the exponential of the square root of its probability”
Those are valid definitions of two different random variables - not a general definition of “random variable”.