What is being said in this example of an infinite sample space? In the notes I'm working through we're given the following example of an infinite sample space:
"Experiment is choose a point in $\mathbb{R}$ (or $\mathbb{R}^n$ for $n \geq2$ ) from a probability distribution function $F:\mathbb{R} \to \mathbb{R}$ (where F is increasing, right continuous)
Then $\Omega=\mathbb{R}$."
I can see why the sample space is $\mathbb{R}$ but what does it mean to choose a point in $\mathbb{R}$ "from a probability distribution function F" ? Is this just shorthand for saying that we choose an element in the domain of F?
 A: Choosing a point from a probability distribution $F$ is called sampling from a distribution. Let the point you choose be denoted by $X$. Before you choose a point, $X$ is unknown; after you choose a point, $X$ realizes a specific value, say $X = x$. 
Therefore, choosing a point is a probability experiment, $X$ is a random variable with distribution $F$. 
Before you choose a point, you can talk about probability  of choosing a value $\leq 4$, i.e. $P(X \leq 4)$. Then, $P(X \leq 4) = F(4)$. 
If you carry out the experiment (choosing a point) several times, you will get samples from $F$, call them $x_1, x_2, ..., x_n$. The mean of the samples $\frac{1}{n} \sum \limits_{i=1}^n x_i$ will approximate the actual mean of the distribution $F$. 
In fact, it can happen you don't know the distribution $F$ but only have samples $x_1, x_2, ..., x_n$ from the unknown distribution $F$. You can use these samples to approximate $F$ at any given value. Say you want to approximate $F(y)$ where $y \in \mathbb{R}$. $$F(y) = \frac{1}{n} \sum_{i=1}^n \mathbb{I}\{ x_i \leq y \}$$
where $\mathbb{I}$ is the indicator funtion. 
