Integration notation

The classical notation for integrals is:
$$\int_a^b f(x)\,dx$$
This has much to recommend it, but I prefer alternatives.

Ultimately, you want the following data when specifying an integral:
- function
- domain
- measure

This is a bit of a mouthful for novices;
further, the measure is generally implicit (the usual Lebesgue
measure).

I prefer to start with:
$$\int_{x=0}^1 f(x)$$
and then generalize to
$$\int_A f(x)$$
with the above being the special case
$$\int_{[0,1]} f(x)$$

[you can even drop the $(x)$; I prefer to drop it,
 but it's a useful reminder that $f$ is a function]

The notation
$$\int_{x=0}^1 f(x)$$
makes it more explicit that $x$ is changing than 
$$\int_0^1 f(x)\,dx$$
this is especially useful when doing double/triple integrals.

Further, 
$$\int_{x=0}^1 f(x)$$
is intentionally very similar to
$$\sum_{i=1}^n a_i$$

The notation
$$\int_A f(x)$$
really emphasizes that the geometry of the base -doesn't matter-:
it's just a measure space.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Measures

Note that the "domain/measure" distinction isn't meaningful;
instead of specifying $A \subset X$, and a measure $\mu$,
you can just look at $\mu \cdot \chi_A$.

This is particularly important as once you've specified a base space
$X$ (or rather, a $\sigma$-algebra ``of measureable sets''), you have
a bilinear function taking in measures and functions and spitting out
the integral.

Notationally, once measures have been discussed,
then one can incorporate a measure into the notation:
$$\int_A f(x) \,d\mu$$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
A key point is to think about how much of the notation isn't
meaningful, or is just ``a reminder''.
$$\int_0^1 f$$
is a fine abbreviation for
$$\int_0^1 f(x)\,dx$$
and the $(x)\,dx$ is just a reminder.
