Accounting for membership in set, but also quantity in event

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?

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 $$\Omega$$.

• thanks so much for the quick response, this was spot on and super helpful! Apr 15, 2019 at 14:25
• actually would we still need the binary sample space (or can we remove it)? the real number space already contains all the information, right? Apr 15, 2019 at 14:36
• 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.
– Ben
Apr 15, 2019 at 23:26