A formal definition of a "measure of association" I've been trying to come up with a formal definition for a 'measure of association'. An intuitive definition might be something along the lines of 'a function that tells you about the existence or strength of dependence among a collection of random variables'.
I've constructed the following definition with this intuitive notion of association. Notice that I use an implication, rather than a biconditional. This is to allow for a function to tell us about specific types of association, rather than dependence in general.

Given a suitable probability space $(\Omega, \mathcal{F}, P)$ with real-valued random variables $\{X_j{(\omega}) | \omega \in \Omega \}_{j=1}^{n}$, a measure of association of order n is a function $f:\mathbb{R}^n \mapsto \mathbb{R}$ such that $\perp\!\!\!\!\perp \left( X_1, \cdots, X_n \right) \implies f \left( X_1, \cdots, X_n  \right) = 0$.

However, it comes up a bit short. This definition doesn't really involve any notion of quantifying the strength of association. I've been mulling over the idea that with sufficient smoothness that perhaps some expression in terms of derivatives would be possible. In a comment below, @whuber nicely summarizes my dissatisfaction with this definition:

It would be more accurate to characterize your definition as an indicator of association. To be a "measure," it ought to change monotonically with some property of "association." The issue revolves around what might constitute a property one would characterize as quantifying some aspect of "association." The main difficulty is that "dependence among variables" is a rich and complex thing that is inadequately characterized by any single scalar-valued function. AFAIK, there is no axiomatization of such things.

How can this definition be revised to include functions that quantify the strength of association?

I think about this a little differently than I used to, and will respond to myself here. kjetil's answer is still the best researched answer here and Ben's answer is thought provoking. The following are just my idiosyncratic brain droppings.
Beginning with a definition of statistical independence
$$F_{X_1, \cdots, X_n}(x_1, \cdots, x_n) = \prod_{j=1}^n F_{X_j}(x_j)$$
where $F_{X_1, \cdots, X_n}$ is the joint cumulative distribution function (CDF) and $F_{X_j}$ is the marginal CDF for the $j$th variable.
Not all variables are statistically independent, which prompted me to define the notion of an independence gap to be
$$\phi_{X_1, \cdots, X_n}(x_1, \cdots, x_n) \triangleq  F_{X_1, \cdots, X_n}(x_1, \cdots, x_n) - \prod_{j=1}^n F_{X_j}(x_j)$$
by simply by taking the difference of the product of the marginals from the joint.
When $\phi > 0$ things are happening more together than they do apart, and similarly when $\phi < 0$ things are happening less together than they do apart. So to speak, anyway. The former to me seems intuitively to match the term "association", and the latter "disassociation".
Any function $g(x_1, \cdots, x_n)$ which is (co)monotonic with the independence gap is a measure of association, and any function $h(x_1, \cdots, x_n)$ which is antimonotonic with the independence gap is a measure of disassociation.
