I find the following remark by Rota interesting:

Insofar as the Laplace transform goes, two radically different uses of the word “function” are dangerously confused with each other. The first is the ordinary notion of function as a something that has a graph. The second is the radically different notion of function as density, whether mass density or probability density. For the sake of the argument, let us agree to call this second kind of function “density function.” Professional mathematicians have avoided facing up to density functions by a variety of escapes, such as Stieltjes integrals, measures, etc. But the fact is that the current notation for density functions in physics and engineering is provably superior, and we had better face up to it squarely

I've suggested to mathematicians that the notation

  ⌠             
  | f(x) dμ(x)
  ⌡             

in measure theory where μ is a measure should be replaced with

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

where the measure μ is treated as a density function. What I've found is a mixture of either not understanding the logic, or the response "What's the point?" So it's interesting seeing Gian-Carlo Rota suggest the same idea AFAICT.

As a bonus, the Radon-Nikodym "derivative" would be written as μ/λ instead of dμ/dλ. The RN derivative then doesn't appear like a derivative - it looks simply like division.