TITLE: Defining Measures on a Manifold: How To
QUESTION [10 upvotes]: 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?

REPLY [11 votes]: 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 $0$.