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Math Help - consistent property

  1. #1
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    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?

    Thank you in advance

    Regards
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  2. #2
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    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.
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  3. #3
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    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 (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
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  4. #4
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    Can nobody help me?
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