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Everyone:

Given smooth manifolds $M,N$ ($m$- and $n$- manifolds respectively) Sard's theorem says that for $f:M \to N$ in $C^k$ ; $k \geq 1$, the image of the set of critical points of $f$ in $M$ --points in $M$ where the Jacobian has rank $< m$ --has measure zero in N:

http://en.wikipedia.org/wiki/Sard%27s_theorem

Now, I know what it means for a set to have measure zero in the case of $M, N$ being Euclidean spaces, but I am not clear on how we define measures on manifolds; I know we may pull back different types of objects from $\mathbb R^n$ into a manifold, like $k$-forms, etc., but I don't know if/how one pullsback a measure from $\mathbb R^n$, since I am not even clear on what kind of object a measure is. Is it just a $0$-form, i.e., just a function? If so, can we define a measure globally on a manifold, or do we just pullback separately for each chart?

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    @t.b: thanks; I will read Willie's answer, but, just for now, let me follow up with Zhen Lin: yes I know some measure theory; generalized measure spaces and sigma algebras. I guess I would/could then define a sigma algebra on my manifold, maybe the sigma-algebra generated by the open sets in the manifold (with its standard metric topology given by the interinsic metric). But then what? Do I use some sort of pullback metric by the chart map?2011-11-05

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Having measure zero is something that can be defined without actually defining a measure on the manifold. A subset $A$ of $N$ can be defined to have measure zero if for every chart $\phi:U\to\mathbb R^n$ of $N$, $\phi(U\cap A)$ has measure zero in $\mathbb R^n$.

You would want this to be something that is invariant under diffeomorphism. That is the case because smooth maps between subsets of $\mathbb R^n$ preserve the property of having Lebesgue measure $0$.