Talking to students in different areas and taking different classes in math, physics, electrical engineering I have been struck by the differing amounts of rigor in use.
I know little about economics but I'm told that they use measure theory and prove rigorous theorems. Yet in physics, perhaps the most mathematical of the sciences, they solve differential equations and whatever else they do without having taken real analysis (usually, as far as I know). Is this for purely historical reasons or do economists really have more need for existence theorems for example?
Another question I have is about probability. It is taught in different ways to undergrads (with talk of different formulae for discrete and random variables) and graduate students (measure theoretically). Is there some great reason for this? I guess to do Brownian motion and similar you need measure theory (is this true?) but does measure theory give you better theorems or proofs than the undergrad approach where both are possible? In information theory it seems they are happy with undergrad level stochastic processes in their textbooks. Would they be improved by using measure theory?
So, as mathematicians (or perhaps physicists, economists, information theorists etc if that is who you are) do you think the different fields have it right?