# consistent property

• Dec 24th 2010, 02:48 PM
Sogan
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) $\subset \mathbb{R}^m$ has measure zero for all coordinate system x $\in$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?

Regards
• Dec 24th 2010, 05:45 PM
xxp9
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.
• Dec 25th 2010, 03:19 AM
Sogan

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 $(U_i,x_i), (V_j,y_j)$ be charts, s.t. $A \subset \bigcup U_i,$ and $A\subset \bigcup V_j$
then A has measure zero with respect to $(U_i,x_i)$(, i.e. $x_i(U_i \cap A)$ has measure zero for all i) iff it has measure zero w.r.t. $(V_j,y_j)$.

But if we proove this, then it is automatically obvious that it suffices to check this property for one atlas, isn't it?

Regards
• Dec 27th 2010, 04:10 PM
Sogan
Can nobody help me?