Often during informal discussion with colleagues, the following pattern emerges when we are stuck trying to prove a theorem about $x \in X$.

A: "let's assume this hypothesis $H$ on $x$"
B: "most elements of $X$ do not satisfy $H$".
A: (censored to protect the innocent).

Of course this makes sense when one restricts to a set $X$ equipped with a sigma algebra and measure, but what happens when you want to measure much larger things? Here are some examples off-hand statements that sound reasonable:

1. Most topological spaces are not Hausdorff.
2. Most categories contain a non-identity morphism.

Is there any logical framework that makes this sort of informal statement precise?

I suppose that one of the most basic statements of this type would be "Most sets are infinite", so one sane approach would involve some correspondence of the form

$$\Omega: \text{[Classes or Large Categories]} \to \text{[Cardinals or Ordinals]}$$

that copies the basic properties of a measure on $\mathbb{R}$. There has been some prior discussion here on extending the range of $\Omega$, but (as you can see from the informal statements above) my primary interest is in extending the domain beyond Sets. Sorry about the fuzzy question, but hopefully there are others who share my experience and want to know.