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Say we have a sample space $$\Omega_1 = \{\text{"alpha"},\text{"beta"},\text{"gamma"},\text{"delta"}\}$$ if we only care about the (binary) membership in set in each event, the event space would be the power set of $\Omega_1$: $$\mathcal{F_1} = \{\emptyset, \{\text{"alpha"}\},\{\text{"beta"}\},...,\{\text{"alpha"},\text{"beta"},\text{"gamma"}\}, \Omega_1\}$$

But what if, in each event, I want to also include the (real-valued) "quantity" of each element in set, in addition to the binary membership?

One approach that came to mind is to have a separate, real-valued continuous sample space $\Omega_2 = [0,100]$

Then the joint sample space would be $\Omega_\text{1,2}=\Omega_1\times\Omega_2$

And an example event will look something like: $\{\text{"alpha"}, 32.8,\text{"beta"},43.6,\text{"gamma"},7.21\}$

Does this approach make sense? Or are there better ways to do this?

And what would be the appropriate notation?

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What you want here can be accomplished by using the sample space:

$$\Omega = \{ 0,1 \}^4 \times \mathbb{R}^4.$$

This sample space allows for four binary indicators and four corresponding real values. The event space $\mathscr{G}$ would then consist of the class of all Borel sets on $\Omega$.

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  • $\begingroup$ thanks so much for the quick response, this was spot on and super helpful! $\endgroup$ – jytoronto Apr 15 '19 at 14:25
  • $\begingroup$ actually would we still need the binary sample space (or can we remove it)? the real number space already contains all the information, right? $\endgroup$ – jytoronto Apr 15 '19 at 14:36
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    $\begingroup$ It depends - your specification was to have a binary membership separate to the real measurement, but you could certainly fold it in if you want. $\endgroup$ – Ben Apr 15 '19 at 23:26

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