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Warning: I don't know anything about set theory so I wouldn't really know how to spot an existing answer if it were around.

Suppose I want to model some economic good or product. I would like to classify the good by the attributes it has. For example, maybe I want to model fruits and I start thinking about the attributes that fruits have:

  • Attribute 1: Color
  • Attribute 2: Shape
  • Attribute 3: Nutritional information

I look at this model and I think: what about the different ways that we can describe color, like hue, saturation, etc? What about all the shapes that fruits could have? What about all the different ways we could break down nutritional information? So I try to dis-aggregate all the attributes as best I can but I keep finding that there are more ways to dis-aggregate. I would like to think of some way to index all the attributes and assign each one a bounded value in $\mathbb{R}$ to describe the product like $Fruit=\sum{a_i\hat x_i}$ but this seems like it's a problem because I am not convinced that I can even index all the ways you might describe a fruit.

I could just give up and say, well, let's just capture the stuff that's really important to what I am trying to study. But suppose that I was not really even all that interested in the specific attributes - just more of the relationships between different fruits $a_i$ distributions - really I just want some canonical way to describe fruits.

Is it in any way reasonable to think that I could place the uncountably many attributes that describe the fruit on some real interval and consider the bounded function $a(x)$ instead? Then, if I wanted to compare two fruits $f_1$ and $f_2$, I could just compare the shapes of $f_1(x)$ and $f_2(x)$. One problem seems that even if you could place the attributes on an interval, it doesn't really make sense to think of $a(x)$ as being necessarily continuous - but if the goal is just to compare two functions there could be some ways to get around that.

It seems very reasonable to place something like "number of coins" on the real line, but pretty absurd to place attributes of objects on the real line. What exactly is the difference and how would you go about describing something like attributes with coefficients in a meaningful way?

(The reason I am asking is because I think I have seen something roughly like this before in a few economics papers.)

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There are models that achieve basically what you are asking for. In general equilibrium theory with differentiated commodities, commodity bundles are often modeled as signed measures of bounded variation on a compact metric space of characteristics.

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