Confusion about measure theoretic approach of probability

Question

Consider the case when someone starts with a random number $X$, then generates consecutive numbers by the recurrence relation $X = X + 1$.

The measure-theoretic approach of probability states that if $X$ is a random variable on $\Omega$, and $f: \mathbb{R} \rightarrow \mathbb{R}$ is a Borel-measurable function, then $Y=f(X)$ is also a random variable on $\Omega$.

Now according to the proposed case, $f(x) = x + 1$.

So $Y_1 = f(X)$, $Y_{i+1} = f(Y_i), \forall i > 0$

Now according to the theory all elements of the sequence $Y_i$ are a random variables, therefore the entire sequence is random, but heavily correlated. This doesn't make much sense to me, since the sequence is easily predictable and obviously not random.

Is it yet true? If yes, then my definition of randomness if wrong and I would like to know what it is actually and how could we say that a sequence is random if it is (heavily) correlated?

Answer 1

A random variable is defined as a measurable function from one space $(\Omega, F)$ to another, say, $(\Omega', G)$.

So, even if you define $Y(w)=1$ for all $w\in\Omega$, Y is still a random variable because Y is a measurable function from $\Omega$ to R (check definition of measurable function). You can choose almost any $F$ and $G$ in this case. Let's just say, $G$ is Borel set, and F is any $\sigma$-field you want.

If you ask why we didn't exclude your case in definition, well, excluding such cases is certainly possible but it really does no good and makes the theory unnecessarily complex. So we choose to include degenerated probability you mentioned as 'random variables' anyway.