My professor of statistics based his lesson notations on H. Georgii's work 'Stochastics'. The sample set is thereby notated as $\chi$, instead of the usual $\Omega$. I don't really understand the reason for this. At page 190, Georggi writes:
The notation $\chi$ instead of $\Omega$ is based on the idea that the observation is given by a random variable $X\colon\Omega\to\chi$, where $\Omega$ yields a detailed description of the randomness, whereas $\chi$ contains only the actually observable outcomes. However, since only the distribution of $X$ (rather than $X$ itself) plays a role, $\Omega$ does not appear explicitly here.
Although this fragment is obviously ment as a clarification, I don't fully get it. Can you explain it a little bit better for me, please?
It turned out that Georgii's notation is quite intuitive. He doesn't use $\Omega$ to underline the difference between the actual sample set and the observations that statisticians use.
$\Omega$. This is the 'real world', the place where observable things happen (e.g. a dice is thrown and the upper side contains six nice, black dots). One easily sees that it isn't necessary to pass all the available details (the object, the amount and color of the dice's dots...) to mathematicians who have to work with this information.
$\chi$. Therefore, the statistician is only given the relevant data (e.g. the amount of dots). These obsevations are actually a kind of mapping from $\Omega$ to $\chi$. This function maps the event to concrete data that can be used to do statistics. (e.g. one doesn't work with "a dice has just been thrown and eventually, hooray, the side with six black dots lay at the top", one simply uses "6".)