
consistent property
Hello,
i try to solve this exercise, but i don't understand it really:
"A subset A of a manifold M has measure zero if x(A) $\displaystyle \subset \mathbb{R}^m$ has measure zero for all coordinate system x $\displaystyle \in $A.
Show that this definition is consistent and that is suffices to check this property for any atlas."
My question is: what does "consistent" mean? Have you an idea?
Thank you in advance
Regards

to define measures on a manifold, one just needs to carry the definition from a coordinate to the manifold. This is what the statement is trying to express.
"Consistent" means it is welldefined. That is, if A has measure zero in one coordinate system, it must has measure zero on any other coordinates.

Thank you for your help!
I think i have understand what you mean. But i don't see the differece between the first statement and the second one.
If the definition is consistent, isn't it the same to proof the secont statement, that is:
"...and that is suffices to check this property for any atlas."
1)The first statement is:
Let $\displaystyle (U_i,x_i), (V_j,y_j)$ be charts, s.t. $\displaystyle A \subset \bigcup U_i,$ and $\displaystyle A\subset \bigcup V_j$
then A has measure zero with respect to $\displaystyle (U_i,x_i)$(, i.e. $\displaystyle x_i(U_i \cap A)$ has measure zero for all i) iff it has measure zero w.r.t. $\displaystyle (V_j,y_j)$.
But if we proove this, then it is automatically obvious that it suffices to check this property for one atlas, isn't it?
Regards
