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?
On
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).
There are homeomorphisms (see Henning Makholm's very good answer) that take a set of zero measure to a set of positive measure (where again measure is taken w.r.t. some local charts), so without modification "measure zero" is not well-defined for topological manifolds. (This pathology is not possible, however, on smooth manifolds.)
On the other hand, Sullivan proved that any topological manifold of dimension $\neq 4$ has a unique compatible Lipschitz structure, that is, a collection of charts whose transition functions are Lipschitz. Lipschitz-ness precludes the sort of pathology described about, so restricting our construction to such a collection yields a well-defined notion of measure zero for topological manifolds (again, except in dimension $4$).