Precise definition of the support of a random variable $\newcommand{\F}{\mathcal{F}} \newcommand{\powset}[1]{\mathcal{P}(#1)}$
I am reading lecture notes which contradict my understanding of random variables. Suppose we have a probability space $(\Omega, \mathcal{F}, Pr)$, where 


$\Omega$ is the set of outcomes
$\F \subseteq \powset{\Omega}$ is the collection of events, a $\sigma$-algebra
$\Pr:\Omega\to[0,1]$ is the mapping outcomes to their probabilities.


If we take the standard definition of a random variable $X$, it is actually a function from the sample space to real values, i.e. $X:\Omega \to \mathbb{R}$.

What now confuses me is the precise definition of the term support. 

According to Wikipedia:


  the support of a function is the set of points where the function is
  not zero valued.


Now, applying this definition to our random variable $X$, these lectures notes say:


  Random Variables – A random variable is a real valued function defined
  on the sample space of an experiment. Associated with each random 
  variable is a probability density function (pdf) for the random
  variable. The  sample space is also called the support of a random
  variable.


I am not entirely convinced with the line the sample space is also callled the support of a random variable. 

Why would $\Omega$ be the support of $X$? What if the random variable $X$ so happened to map some element $\omega \in \Omega$ to the real number $0$, then that element would not be in the support?

What is even more confusing is, when we talk about support, do we mean that of $X$ or that of the distribution function $\Pr$? 

This answer says that:


  It is more accurate to speak of the support of the distribution than
  that of the support of the random variable.


Do we interpret the support to be


the set of outcomes in $\Omega$ which have a non-zero probability, 
the set of values that $X$ can take with non-zero probability?


I think being precise is important, although my literature does not seem very rigorous.
I am not entirely convinced with the line the sample space is also called the support of a random variable 


That looks quite wrong to me.


  What is even more confusing is, when we talk about support, do we mean that of $X$ or that of the distribution function $Pr$?


In rather informal terms, the "support" of a random variable $X$ is defined as the support (in the function sense) of the density function $f_X(x)$. 

I say, in rather informal terms, because the density function is a quite intuitive and practical concept for dealing with probabilities, but no so much when speaking of probability in general and formal terms. For one thing, it's not a proper function for "discrete distributions" (again, a practical but loose concept). 

In more formal/strict terms, the comment of Stefan fits the bill.

Do we interpret the support to be

- the set of outcomes in Ω which have a non-zero probability,
- the set of values that X can take with non-zero probability?


Neither, actually. Consider a random variable that has a uniform density in $[0,1]$, with $\Omega = \mathbb{R}$.
Then the support is the full interval $[0,1]$ - which is a subset of $\Omega$. But, then, of course, say $x=1/2$ belongs to the support. But the probability that $X$ takes this value is zero.
A random variable is defined as a function that maps outcomes to numerical quantities, typically real numbers, i.e., X: Ω ↦ R. 

The “domain of a random variable” is the set of possible outcomes. In the case of the coin, there are only two possible outcomes, namely heads or tails. Since one of these outcomes must occur, either the event that the coin lands heads or the event that the coin lands tails must have non-zero probability. For an unbiased coin p(H)=p(T)=1/2.

Consider another random variable Y, be the number of heads. So toss a coin twice produce y = 0, 1 and 2 with p(Y) = ¼, ½ and ¼ respectively. 

The “support of a real-valued function f ” is the subset of the domain containing those elements which are not mapped to zero.  Let f(x) be the p.d.f. of normal distribution with support x∈R , so that f(x)≠0.