From: ed...@math.ohio-state.edu (G. A. Edgar)
Subject: Re: Use of the notation "d" in measure theory
Date: 1998/08/24
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> It is always used in front of measures.

Not always.  There are those who put it inside, like this:
   \int_A f(x) \mu(dx)
[Here, we think of dx as a "small" set.  We evaluate f at a point in
the set, multiply by the measure of the set, and add up lots
of these forming a partition of A.  The integral is the limit of such things,
under refinement of the partition.  Such a limit really produces the
Lebesgue integral.]

....

> Appendix:
> 
> Here is a logical development of integration theory using my notation:
> Define measurable spaces, measurable functions, and *signed* measures.
> Then define the measure fm by first defining it
> when f is a characteristic function, then when f is a step function, etc,
> so that you develop fm(A) the same way you normally develop 
> \int_A f m (previously called "int_A f dm").
> There is a theorem here; that (gf)m = g(fm).
> Now introduce, purely as a matter of *notation*, "\int_A m"
> to mean m(A) for any measure m, including one of the form fn.

I have seen a development something like this.  But in fact it was used to
define \int_A f m, where m is a set function more general than a measure...
semigroup-valued, or something.

...

Here is a question for your notation:  In the theory of Markov
chains, we often talk about a "transition probability", which
is a function of the form p(x,A), where for each fixed set A
it is a measurable function of x, and each fixed point x it is
a measure as a function of A.  We want to use integrals like:

    \int_A  g(y) p(x,dy)

What is your notation for this?
-- 
Gerald A. Edgar                   ed...@math.ohio-state.edu