TITLE: Is there a well-defined notion of measure zero on topological manifolds?
QUESTION [5 upvotes]: We extend the concept of measure zero on manifolds by local parameterization.
but in this definition we have to check if it is true for every parametrization.
In Guillemin's Differential Topology this is stated for differentiable manifolds.
But, what if we just have a topological manifold (i.e., every point has an open neighborhood homeomorphic to some Euclidean space)?  Is measure zero still a well-defined concept in this setting?

REPLY [9 votes]: No. Just for good old $\mathbb R$, there is a homeomorphism $[0,1]\to[0,1]$ which takes the middle-thirds Cantor set (of measure $0$) to the "fat" Smith-Volterra-Cantor set (of measure $1/2$).
(The excluded intervals in the two constructions, including their endpoints, correspond naturally to each other -- simply map each of them to its counterpart interpolating linearly between the endpoints. Then continuity fixes the image of those points of the Cantor set that are not interval endpoints).