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) has measure zero for all coordinate system x 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
December 24th 2010, 04:45 PM
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 well-defined. That is, if A has measure zero in one coordinate system, it must has measure zero on any other coordinates.
December 25th 2010, 02:19 AM
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 be charts, s.t. and
then A has measure zero with respect to (, i.e. has measure zero for all i) iff it has measure zero w.r.t. .
But if we proove this, then it is automatically obvious that it suffices to check this property for one atlas, isn't it?