# Mixing category and measure

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 $$\rho(\bigcup_{i\in\mathbb{N}}C_i)=1>sup\{\rho(C_i):i\in\mathbb{N}\}$$.

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.