Recall that the Lebesgue measure $\mu$ is often used to show how ''insignificant'' a subset is compared to $\mathbb{R}$. For example, $\mu(\mathbb{Q})=0$ even though $\mathbb{Q}$ has an infinite number of members.
I would like to show a similar thing for probability measures.
Specifically, consider two sets $A$ and $B$, their Cartesian product $A \times B$, and a sigma algebra over the Cartesian product $\sigma(A \times B)$. Let $U$ denote the set of all probability measures on $A \times B$. Using some ''measure of probability measures'' $\delta$, I would like to show that the subset $V$ (of $U$) consisting of all independent probability measures on $A \times B$ is ''insignificant'' compared to $U$. That is, $\delta(V) = 0$. Here, $V$ is composed of the probability measures which satisfy $P(C_A, C_B)=P(C_A)P(C_B)$ for any $C_A \in \sigma(A), C_B \in \sigma(B)$ and $ C_A, C_B \in \sigma(A \times B)$.
Does anyone have any ideas on how such a ''measure of probability measures'' $\delta$ would be defined?
One way would be using random measures: https://en.wikipedia.org/wiki/Random_measure. Then an insignificant set of probability measures is a set with probability zero (this does not have to be the empty set).
Obviously this notion of insignificance depends on the chosen distribution of the random measure (i.e. the probability measure on the space of probability measures), although the notion will coincide for distributions (probability measures on the space of probability measures) which are absolutely continuous with respect to each other, although that is more or less by definition. https://en.wikipedia.org/wiki/Absolute_continuity#Absolute_continuity_of_measures
Needless to say any way to do so requires a lot of measure theory machinery.
You say you want to show that the space of probability measures on the product space which satisfy the independence property is "trivial". If we have any measure of probability measure which does not assign non-zero mass to any single individual probability measure (i.e. to any single point of the space of probability measures), then your desired conclusion will follow from the fact that the "product measure", which corresponds to independent probability measures is a single point (when we fix the probability measures on the factor spaces $A$ and $B$).
This is because the product measure is unique https://en.wikipedia.org/wiki/Product_measure, since the probability measures on $A$ and $B$, being finite measures, are in particular $\sigma-$finite.
I think a better argument that such measures are "insignificant", however, is whenever we fix two marginal distributions on $A$ and $B$, there are uncountably many possible probability measures on the product space $A \times B$ (provided of course that $A$ and $B$ are reasonably complicated, which is waht I am assuming), whereas the product distribution corresponding to independence is a single point. And clearly any reasonable notion of "significance" will designate a single point in a set of uncountable cardinality as insignificant.
Another way to show that there are uncountably many possible probability measures on the product space is to think in terms of correlation coefficients for the two marginal distributions depending on the joint distribution. One can construct at least one joint distribution on $A \times B$ such that the marginal distributions have correlation coefficient $\rho \in [-1,0) \cup (0,1]$, and then even of those joint distributions such that the two marginals are uncorrelated, not even all of them will be independent (in fact, only one of them will be). Again, all of this is assuming that $A$ and $B$ as well as the probability measures on them are reasonably complicated enough; as an example I am thinking of probability measures absolutely continuous with respect to Lebesgue measure on $\mathbb{R}$.
Without more details on your part (which ultimately might be more a matter of taste or convenience than true mathematical insight), I don't think I am able to say anything more specific.