@PaceNielsen Oh, I see what you're asking. No, in this case you cannot take intersection in ZFC. If you could define an intersection, then you would run in inconsistency.

@PaceNielsen I think the message is: whenever you can actually right down, for a general $n$, what the class $A_n$ is, then your family of classes is defined uniformly, and you really look at sections of one class. So you can freely intersect them.

@PaceNielsen Why is $\phi(R,n)$ indexed by meta-natural number? I mean, you definitely can define your class this way, e.g., you can define \phi(R,3) by saying that you consider matrices with distinct entries a,b,c,d,e,f,g,h,i. But you definitely don't have to, and you didn't when formulating your definition. If you really insisted on using meta-natural numbers in your definition, then you couldn't have phrase your example correctly with a formula in ZFC (which you clearly did)

That's a fair point. Both as a student and as a teacher, I first had full presentation of Lebesgue measure, and only then of Lebesgue integral, so I missed that point. Do you know whether the route you sketched is actually followed somewhere or is it just a hypothetical scenario? I believe that probabilists might want it this way.

@PiotrHajlasz On the other hand, you probably want to prove how measure behaves under affine transformations anyway, after which your trick with addition comes virtually for free.

@mathreadler You are right, I edited the question, to remove possible uncertainty as to the intended reading.

@DaveLRenfro I think it might be reasonable to post it as an answer.

@Basj Just for a context: I asked this question, since I was grading an exam, where in one exercise clearly students would have much easier life with the area-under-the graph definition of the integral and where clearly many people were confused with some aspects of the theory..

@Basj Yes, it takes sometime to develop measure. From the distance, it might seem that developing integral from that point is immediate. From my very recent teaching experience, when I taught this stuff for the first time: it's really not. You have to add a whole new layer of the theory, prove some facts about measurable functions, and only then you are allowed to integrate. My question was really whether this process cannot be streamlined significantly..