Sample: don't confuse measurements with actual values? In Wikipedia's article on Sample there is the following remark:

''Note that a sample of random variables (i.e. a set of measurable functions) must not be confused with the realizations of these variables (which are the values that these random variables take). In other words, $X_i$ is a function representing the measurement at the $i$-th experiment and $x_i = X_i(ω)$ is the value we actually get when making the measurement.''

I'm afraid I don't understand this passage, can anybody please explain the point?
Let's assume that there are infinitely many humans and that their IQ has a normal distribution with mean 100 and variance 15. Here is a possible experiment:


  grab 10 people downtown and measure their IQ


As a mathematical model, we accept the setting that there is a probability space $\Omega$ and "grab human number i" is modelled by a "random variable" $X_i$ which is a measurable function
$$
X_i: \; \Omega \to \mathbb{R}
$$ 
such that the $X_i$ are independent and have a gaussian aka normal distribution.

When you go out and acutally perfom this experiment, you'll get ten humans and ten values (real numbers) for theiy IQ's. Let's call them $IQ_i$. With regard to our mathematical model, this means that for every human you convinced to do the IQ-Test there is a corresponding $\omega \in \Omega$, each of which is called an event, such that 
$$
IQ_i = X_i(\omega)
$$

When you get out again and perform this experiment again, you can accept the same random variables $X_i$ as a model for your experiment, but you'll encounter different humans, get different results from the IQ-Tests, which corresponds to a different $\omega &#39; \in \Omega$.

Or, to put it shortly as Didier Piau did: The description of the experiment entails as a mathematical model the random variables  aka measurable functions $X_i$; but everytime when you actually perform the experiment, it will result in a tuple of values of these functions.
When a random variable takes a value, you cannot always explicitly know this exact value, you can only measure it, i.e. use a function of the result which outputs you the measuring. For instance, would you be sampling real numbers in $[0,1]$ and measuring them only up to 3 digits of precision, if your first random variable $X_1$ gets value $\pi/4$ for instance, you could only say $X_1 = 0.785$, since the measurable function $X_1$ maps any $\omega$ in the interval $(0.7845,0.7855]$ to $0.785$.

In other words, $\omega$ is the "actual value", if we could say this, of the random variable and $X_1$ is a way to measure the actual value by letting it become a measurable function.

Hope that helps,