On a complete metric space the collection of meager and comeager sets form a $\sigma$-algebra. There is a 'natural' measure you can put on this $\sigma$-algebra where the measure of a meager set is 0 and the measure of every comeager set is some non-zero value such as 1 or $\infty$. When can such measures be extended to Borel, inner/outer regular, locally finite, and/or Radon measures? Can it be done in such a way that the only null sets are meager sets?

## 1 Answer

Measures that vanish on all meager sets are called *residual measures*. Measures whose null sets are precisely the meager sets are called *category measures*.

Szpilrajn was the first to show that a separable metric space with no isolated points has no nonzero finite residual Borel measures. The point mass of an isolated point is a nonzero residual measure. This was extended by Marczewski and Sikorski to metric spaces whose weight is less than the first real-valued measurable cardinal. Apparently Flachsmeyer and Lotz removed this large cardinal restriction for metric spaces (it's necessary for more general compact Hausdorff spaces), but I've never seen the proof.

On metric spaces, Oxtoby showed (in the final theorem of that paper) that (finite Baire) category measures exist precisely when every meager set is nowhere dense, or equivalently when the set of isolated points is dense. Then the category measures are precisely those that are positive on isolated points and vanish elsewhere. I don't have it to check, but I think he also treats this material in a more pedagogically sound manner in his book Measure and Category.

There are so many definitions of measures being regular or Radon that disagree with each other in subtle ways that are often mistakingly ignored in the literature, so I'm not sure I can give an exact answer without fixing definitions. In that paper Oxtoby shows that every category measure on a metric space is outer regular on closed sets.

These questions get a lot hairier in more general spaces, but there are more natural examples. If you take the measure algebra of a measure (the Boolean algebra of measurable sets modulo the ideal of null sets), then the measure on the original space extended to the Stone space of the measure algebra is a category measure.