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Let $f:M \to N$ be a smooth map of smooth (second countable) manifolds. The set $C_f \subset M$ of critical points of $f$ is defined to be the set of all $m \in M$ such that the differential $df : T_{m} M \to T_{f(m)}N$ does not have full rank. Then, Sard's theorem states that $f(C_f) \subset N$ has Lebesgue measure $0$.

This is the starting point for both Morse theory as well as the differential approach to topological degree theory. After establishing genericity of the desired type of functions, Sard's theorem is not used again in either Morse or degree theory to the best of my knowledge: it is just a starting point.

Now let $\lambda$ be any measure on $\mathbb{R}$ that is (say) absolutely continuous with respect to Lebesgue measure. This $\lambda$ (using the product measure on $\mathbb{R}^{\dim(N)}$ and a partition of unity) induces a measure $\Lambda$ on $N$. Clearly, the conclusion of Sard's theorem immediately applies to $\Lambda$ as well: $\Lambda[f(C_f)] = 0$ for any smooth $f$ mapping into $N$, by definition of absolute continuity of measures.

It seems that one could attempt to reformulate degree theory or Morse theory for the new measure $\lambda$ instead of the usual Lebesgue measure. I'm not sure what one can conclude from such theories, but I am curious to see if someone has developed a general notion of "Sard measure" which starts with a definition equivalent to:

A measure $\lambda$ in $\mathbb{R}$ is a Sard measure if for any smooth $f:M \to \mathbb{R}$ we have $\lambda[f(C_f)] = 0$.

All measures absolutely continuous with respect to Lebesgue measure are "Sard" by the preceding observation, but:

Are there other examples of non-trivial Sard measures, and is there a generalization of Morse and degree theories for arbitrary Sard measures?

Any pointers to the literature, or an explanation of why such a theory would be completely useless are most welcome.

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All of the applications of Sard's theorem to topology that I can think of use only the much weaker statement that every smooth function has a regular value. I think that to make your notion useful one would need a result that depends on how many regular values one has. Actually, the more I think about it the more I believe that "non-Sard" measures have greater potential since a lot of interesting topology happens near critical points. – Paul Siegel Jul 18 '12 at 3:27
You may be interested in this answer: – Eric Wofsey Mar 8 '13 at 16:37

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