[asdf_snar]

I think I am one of these mathematicians that doesn't understand the logic. How can I write μ(x)dx instead of μ(dx) without risking the confusion that dx is Lebesgue measure? You may have explained this in your other reply, but I don't quite follow.

P.S. Beautiful integral signs.

[ogogmad]

I'm suggesting maybe writing the Lebesgue measure as

  1

so the Lebesgue integral of a function f becomes

  ⌠             
  | f(x)⋅1 dx
  ⌡             

The logic is that the Lebesgue measure is a density which is everywhere equal to 1. Given a measurable space over \mathbb R^n, I think there is only one such measure.

Another example is that δ(x) in

  ⌠             
  | f(x)⋅δ(x) dx
  ⌡             

represents the Dirac measure.

For producing this ASCII art, I use Sympy. I write for instance

  pprint(Integral(f(x) * delta(x)))

[mmmmpancakes]

So what does dx mean in this setting then? If the answer is nothing, then let me suggest simplifying your expression to the following:

  ⌠             
  | f(x)⋅μ(x)
  ⌡   

Now it occurs to me that the only problem with this new notation is that you risk confusing which term is the density (especially if there are multiple greek letters floating around). To clarify this potential confusion I have a solution! Add some notation to indicate which is the density:

  ⌠             
  | f(x)⋅dμ(x)
  ⌡   

Wait...

[politician]

Please excuse my imposition, as I am a humble programmer who spends his days adding and subtracting 1, and not a mathematician.

Yet, this discussion of the confusion and potential confusion of misinterpreting notation strikes me as something that has long (well, in the sense of programming) been solved in my area with type systems and syntax highlighting.

Do mathematicians not have these tools?

[Sharlin]

There is no syntax highlighting on the blackboard.

Math notation is not designed. It has haphazardly evolved over centuries. It is not rigorous even though math itself is (attempts to be) rigorous, it is a language as imperfect as its users. But it does its job well enough.

[ABeeSea]

They do! It’s called abstract algebra and it’s very similar to type theory in a lot of ways.

But to get to the rigorous mathematician definition of manipulating dx and dy, it requires a large amount of the machinery from abstract algebra that’s hard to quickly absorb or explain.

[koningrobot]

Did you mean to write dμ(x) instead of μ(dx)? As a non-measure-literate person, I can understand dμ(x) as the d of density μ(x) evaluated at x. But μ(dx) has μ evaluated at the infinitesimal dx which is very different. Is μ(dx) the correct notation?