Example of non random variable Given that a random variable $X$ defined on a sample space which takes value on the real line is defined as the set of all outcomes such that $X(outcome)\le r$, with $r$ a real number, that belongs to the event space for every $r$, can you provide me with one example (or more) of a non-random variable? 
Please give an example that contradicts the part of the statement relative to the fact that the set belongs to the event space, because it is easy to show that a $X$ not defined on the sample space or that takes values outside the real number set is non-random by definition.
I could not use latex properly to write rigorously the definition, however is the definition of Mood, Introduction to the theory of Statistics. 
 A: $\Omega =\{0,1\}, \mathcal F=\{\emptyset, \Omega\}, X(0)=0,X(1)=1$ is an example since $(X=0) \notin \mathcal F$.
A: Take any non-measurable set $A$ (so $A\in \mathcal P( \Omega )\setminus \mathcal F$) and let $X$ be the indicator function for $A$; i.e.:
$$
X(c) = \begin{cases}
1 & c\in A \\
0 & c \not\in A
\end{cases}
$$
Then the set of all outcomes $c$ such that $X(c)<0.5$ is the complement of $A$, which is non-measurable.
A: The identity function $X$  on $\mathbb{R}$ (i.e. $X(r)=r$ for all $r\in\mathbb{R}$) is not a random variable 
for the set of outcomes (sample space) $\Omega=\mathbb{R}$ and the set of events $\mathcal{F}=$ 
{all (at most) countable subsets of $\mathbb{R}$ and their complements}.
 The peculiarity of this example is that, for all $r\in\mathbb{R}$, $(X=r)$ is an event, yet $(X<r)$ is not.
 So "randomness" of a given $X\colon \Omega\to\mathbb{R}$ depends on the choice of $\mathcal{F}$ which is theoretically 
limited only by the requirement that $\mathcal{F}$ be a $\sigma$-algebra.
To apply probability theory to a real-world situation, one has to choose (explicitly or implicitly) some 
probability space $(\Omega, \mathcal{F}, P)$ including a probability measure $P\colon \mathcal{F}\to P$  (assignment of probabilities to events),
the choice being based upon some real-world data.
 Hypothetically, a hard-hitting researcher of coin tossing could refuse to consider $(X=0)$ 
as an event with a prescribed probability, so they could come to $\mathcal{F}=\{\emptyset, \Omega\}$ 
(mathematically correct, but probably of little use in real-world applications).
 The same experiment of flipping a coin, yet another interpretation 
(an answer to Kolmogorovwannabe's comment on Kavi Rama Murthy's answer).
