Baire Category Theorem in a Smooth Manifold

Question

Let $Z\subset M$ be a set of measure-0 , in a [smooth] manifold $M$.

How does one shows that $M$ \ $Z$ is everywhere dense in $M$, using the Baire category theorem?

And which version of the theorem should one use?

Answer 1

The answer is: one does not use Baire category for this purpose. The Baire category theorem is a topological statement and is agnostic of measure theory. The $\sigma$-ideals of category-1 sets and of Lebesgue null sets are very different in Euclidean spaces (hence on smooth manifolds): the complement of a measure zero set can be a set of 1st category.

Presumably, you define the $\sigma$-ideal of null sets on a manifold as being generated by images of Lebesgue null sets on $\mathbb R^n$ under coordinate maps.

As Niels Diepeveen said, the statement you are proving is equivalent to: "the measure of every nonempty open set is positive". Pick a point in the set and a neighborhood of that point that is contained within a coordinate patch. The image of this neighborhood under a coordinate map is a nonempty open subset of $\mathbb R^n$, hence is not a null set.