For $X\subseteq (0,1)$, let $U_X$ be the intersection of all open sets $V$ such that $X\setminus V$ is meager. Let $\rho(X)=\mu(U_X)$. Intuitively, $\rho$ measures how often a set looks comeager.
Sadly, $\rho$ is pretty badly behaved:
It's not regular, since for $C$ closed we have $\rho(C)>0\iff C$ contains an interval and this prevents any closed subset of $(0,1)\setminus \mathbb{Q}$ from having positive $\rho$-value even though $\rho((0,1)\setminus \mathbb{Q})=1$.
It's not even countably additive on the open sets (contra a claim I made earlier - I had a stupid moment)! Let $(C_i)_{i\in\mathbb{N}}$ be a decreasing sequence of closed subsets of $(0,1)$ which are each finite unions of nontrivial closed intervals and have $$X:=\bigcap_{i\in\mathbb{N}}C_i$$ nowhere dense but non-null (e.g. take the standard approximation of a fat Cantor set). Now consider $X_i=(0,1)\setminus C_i$. $(X_i)_{i\in\mathbb{N}}$ is an increasing sequence of open sets and $\rho(X_i)<1-\mu(X)$ since $U_{X_i}=int(X_i)$ (this is where the "finite unions of closed intervals" bit comes in). But since $X$ is nowhere dense we have $\rho(\bigcup_{i\in\mathbb{N}}C_i)=1>sup\{\rho(C_i):i\in\mathbb{N}\}$.
- Note that this is really a consequence of the existence of comeager null sets.
Despite its poor manners, I'm still interested in $\rho$ for logic-y reasons. Unfortunately, I don't have much analysis background. Before I reinvent the wheel I'd like to ask:
What is $\rho$ actually called, and what's a good source on it?
While its definition is natural enough at first glance, it's plausible to me that $\rho$ is so ugly that it's simply not studied; it's always hard to prove that kind of negative, but in the absence of a more positive answer I'll accept an answer that gives a good argument for $\rho$ being excessively terrible.
