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?

Below are some desiderata that might be useful.  I'm not certain these desiderata will work, but this would be a reasonable starting point for inquiry.  Essentially, you need some kind of property to ensure that the "measure" is well-ordered, specified as an inequality that the measure must satisfy.  Here I have used the idea that adding an independent random vector should "derogate" from the association.
The first property is the one you give in your question, but I have also added some other properties that I think would be useful in a measure of association.  The smoothness property reflects the idea that you want your measure to change continuously when you change the random vector continuously.  The derogation desiderata reflects the idea that adding an independent random vector to your existing random vector should not increase the association, and will decrease association when the added vector is non-degenerate.  If you want your measure of association to have a maximum value, I would also suggest the last assumption.

No association: If the elements of $\mathbf{X} = (X_1,...,X_n)$ are mutually independent then we have $f(\mathbf{X}) = 0$.
Smoothness: Given a random vector $\mathbf{Y} = (Y_1,...,Y_n)$ independent of $\mathbf{X} = (X_1,...,X_n)$, the function $f(\mathbf{X} + \alpha \mathbf{Y})$ is continuous with respect to $\alpha$.
Derogation (weak): Given a random vector $\mathbf{Y} = (Y_1,...,Y_n)$ independent of $\mathbf{X} = (X_1,...,X_n)$ we have $f(\mathbf{X}+\mathbf{Y}) \leqslant f(\mathbf{X})$.
Derogation (strong): Weak derogation applies, and additionally, if $\mathbf{Y} = (Y_1,...,Y_n)$ is non-degenerate (i.e., it does not have a point-mass distribution) we have $f(\mathbf{X}+\mathbf{Y}) < f(\mathbf{X})$.
Maximum association (optional): If all values in $\mathbf{X}$ have a point-mass distribution when we condition on any single value $X_i = x_i$ then we have $f(\mathbf{X}) = 1.$

One other thing you should bear in mind here is that you might need to define the "measure of association" with respect to the distribution of the random vector rather than the random vector itself (though there are other concepts in probability/statistics where we define an operation on a random vector that implicitly uses its distribution).
