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I am really struggling with the notation for this. Below is the slide material followed by what I think it means/where I get stuck.

"Some unknown real world quantity $\Theta$ takes values in $\Omega$. Typically $\Omega$ is some p-dimensional subset of $R^p$.

Let $\mathbb{S} \subset \Omega$. Is $\Theta$ in $\mathbb{S}$?"

At first I considered that $\Omega$ was a range of values and $\Theta$ took values in that, let's say it was height in cm.

$\Omega$ is a dimension (characteristic) in $\mathbb R$ (similar to dimensions in Principal component analysis) so could be eye colour, hair colour, age, height etc.

I am not fully sure about those but even if they are true, what is $\mathbb S$? And how is $\mathbb S$ a set in $\Omega$?

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  • $\begingroup$ To address the questions in your title: Could you explain the sense in which something like "hair color" can be considered a component of a vector of real numbers? $\endgroup$ – whuber Jun 11 '19 at 21:23
  • $\begingroup$ I took it to mean then if we are using hair colour that $\Omega$ would be in the range of {0,1} as a binary for each hair colour. Then the event $\mathbb{S}$ occurs is a 0/1. This is all on the assumption that anything I have said is correct. I have been banging my head against the desk for 2 days nearly. $\endgroup$ – Daniel T Jones Jun 11 '19 at 21:26
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Maybe giving an example will clear things up:

"Typically $\Omega$ is some p-dimensional subset of $R^p$" - e.g. $\Omega$ may be a subset of $R^2$ and represent tuples of height and weight measured in cm and kg

"Some unknown real world quantity $\Theta$ takes values in $\Omega$" - I don't know the exact context of your material, but I guess $\Theta$ here is ment to be a random variable taking values from $\Omega$. So a particular realisation of $\Theta$ is an element from $\Omega$ e.g. a tuple (182 cm, 90 kg). Another realisation of $\Theta$ will be another element from $\Omega$ e.g. (174 cm, 70 kg).

"Let $\mathbb{S} \subset \Omega$." - $\mathbb{S}$ is a subset of $\Omega$. E.g. heights between 170 cm and 190 cm and weights between 80 kg and 100 kg.

Now you can ask a question: "Is $\Theta$ in $\mathbb{S}$?" which means "if $\Theta$ is an element of $\mathbb{S}$?" For $\Theta$ equal (182 cm, 90 kg) and $\mathbb{S}$ as defined above it's true, for $\Theta$ equal (174 cm, 70 kg) it's false. If you know the distribution of $\Theta$ you'll also be able to give a probabilistic answer to this question e.g. "$\Theta$ is in $\mathbb{S}$ for 70% of cases".

So $\mathbb{S}$ is a subset of $\Omega$ in the same way as $\Omega$ is a subset of $R^p$. Your confusion may have came from the fact that $\mathbb{S}$ is a subset of $\Omega$ and not its element as realisations of $\Theta$.

I also think you are trying to be more general than you need to be. If $\Omega$ is some subset of $R^p$ it's easier to think of it as representing some continuous properties like height and weight rather than categorical ones like hair colour. Of course you can do it by binarazing as you wrote in the comment, but this makes things a little bit less clear.

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