[asdf_snar]

Thanks for your reply above.

I don't think the comment was sarcastic or rude. They are pointing out the following inconsistency: you've basically attached "dx" to every integration sign, making the "dx" essentially irrelevant.

Moreover, "dx" does not mean "a small change in x". "dx" is a differential form; it is in particular the "d" operator applied to the function f : x --> x.

As I revisit your comment, I think the point Rota is making about physics notation -- which I _do_ agree with -- is that one should use density functions instead of measures, in general. So, for instance, using the Dirac "density"

\int f(x) \delta(x) dx

instead of

\int f(x) \mu(dx)

where \mu is a point mass at 0. This happens again in the context of stochastic differential equations, where mathematicians shirk away from writing dB_t = xi(t) dt, where xi(t) is "white noise". One can make sense of this in the sense of distributions, and then everything happens in a nice inner product space. Indeed, the physicists are much more competent at actual calculations, and the density representation of things (e.g., in terms of "xi") is very useful for those.