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?