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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?

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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.

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  • $\begingroup$ It seems to me that in measure theory a random variable is equivalent to a Borel measurable function. The question is still how could we say something is random, if that is predictable and/or correlated, or as you showed, a constant value $Y(\omega)=1$? I mean in statistical sense that is not random. The terminology is ambiguous. $\endgroup$ – plasmacel May 26 '17 at 10:28
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    $\begingroup$ Well you can say that Y is 'random variable' according to definition, but Y is actually not random. 'Random variable' is a technical term so you should follow the definition even it does not match your intuition. $\endgroup$ – failedstatistician May 26 '17 at 16:24
  • $\begingroup$ Well, if you ask me, the naming "random variable" is bad. Its pretty silly to use the property "random" in the naming when it doesn't need to satisfy it as a requirement. However it makes more sense now, thanks. You verified my assumptions. $\endgroup$ – plasmacel May 26 '17 at 17:48
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    $\begingroup$ Sounds like an empty debate to me: I do not think there is a general agreement about the meaning of randomness outside measure theory. $\endgroup$ – Xi'an Jun 9 '17 at 19:55

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