# Does there exist a universal random variable which can represent all events in a probability space?

Given a probability space $$(\Omega, F, P)$$, I know that for each event $$A \in F$$, we can define an indicator random variable $$I_A$$, then $$A$$ corresponds to $$I_A=1$$. I am wondering does there exist a "universal" random variable $$X$$, such that $$\forall A \in F, A$$ can be represented as $$a \leq X \leq b$$ for some $$a$$ and $$b \in \mathbb{R}.$$

Moreover, given a probability space, does the set of all random variables have any structure? If the "universal" random variable exists, is it unique and what is its role in this set?

• The set of all random variables is the set of all $\mathfrak F$-measurable functions on $\Omega$ and it has lots of natural, useful structures. The question of the existence of a "universal" variable is quickly and simply addressed by considering two different measurable sets and their complements: soon you will see it's impossible for certain combinations of those sets to correspond to intervals of real numbers. That, unfortunately, doesn't leave one much to write about!
– whuber
Mar 9 at 19:04

The result is simple but it's surprisingly hard to demonstrate rigorously: when such a "universal random variable" $$X$$ exists, its image has at most two elements, which implies the sigma algebra $$\mathfrak F$$ has at most four events.

Let's begin by establishing definitions and notation.

I understand that an event "$$A$$ can be represented as $$a\le X \le b$$" means $$a=a(A)$$ and $$b=b(A)$$ are numbers determined by $$A$$ and $$X$$ and that $$A$$ is the set of all outcomes for which the value of $$X$$ lies between $$a$$ and $$b,$$

$$A = X^{-1}([a,b]) = \{\omega\in\Omega\mid a \le X(\omega)\le b\}.$$

To avoid discussing infinities, compose $$X$$ with the (measurable, strictly increasing) function $$x\to 1/(1 + \exp(-x))$$ to assure the image of $$X$$ is in the interval $$[0,1].$$ This does not change the universal representation property of $$X.$$

Now we can carry out the analysis.

Suppose the image $$X(\Omega)$$ of $$X$$ contains more than two values. Pick three of them and call them $$x \lt y \lt z.$$

Because $$\{y\}$$ is measurable, $$B = X^{-1}(y)$$ is a (nonempty) event. The complement $$B^\prime = \Omega \setminus B$$ also is an event. But

$$X(\omega) = y \in [x,z] \subseteq [a(B^\prime), b(B^\prime)]$$

implies, by virtue of the representation property of $$X,$$ that $$\omega\in B^\prime;$$ that is, $$\omega\notin B.$$ This contradiction of the assumption $$\omega\in B$$ implies the original supposition is false: it is not possible for the image of $$X$$ to contain more than two values, QED.