An Extension of Sard's Theorem*

M I G U E L D E G U Z M / ~ N

A classical form of the theorem of Sard affirms that if g: R" ~ R" is a function x = g(y) in c~ I(W,), then the set of critical values, i.e. the set of points x 0 ~ 6"

such that for some yo , g(yo) = xo and I d e t (O~.g.) 1 = O, is of zero measure. y=YO

This theorem can be considerably extended in the following way. One first gives a general izat ion of the not ion of the Jacobian determinant , .then one

defines correspondingly the critical values and proves that the set of critical values has measure zero. It turns out that one can present the theorem in a very general setting as follows.

Let I/ ~ )' = R" be an open set. Let X be another set and p an exter ior mea- sure defined on the subsets of X. Let g: I/ --, X be an arb i t ra ry function. A point y e I/ will be called critical point of g if there is a sequence {Qk(Y)] of open cubic intervals centered at v and cont, 'acting to y such that

p(g((2~(v)) ~ O.

Here IQ[ is the Lebesgue measure of Q.

For the p roof of the theorem we want to present we shall make use of the following covering l emma whose .p roof can be found in [1].

LEMM~. Let S be a bounded set of ,R". For each v ~ S an open cubic interval

Q(.v) centered at v is given. Then one can c-"hoose from among (Q(.v))y~s a se-

quence {Qk] o f such cubes so that

a) S c [ J O g ; k

*Recebido pela SEM em 29 de agosto de 1972.

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b) EZ0~(x) < 0(n) .for ererr x ~ ?~".

Here Ze is the characteristic fimction of P and O(n) is a constant that onh' de-

pends on the dimension.

THEO!~.EM. Let g: V -, X be as in the precediJu I definition. I f C is the set o f

critical points, then p(g(C)) = O, i.e. the set of critical t~alues has p measure zero.

PROOF. We first take a bounded subset S of C and prove p(g(S)) = 0. Since

C = U Sk, Sk bounded, we then have k

p(g(C)) = p(g(U Sk)) <-- P(U g(Sk)) <-- ~ p(g(S~)) = O. k k k

Let then S be an arbi t rary bounded subset of C. Let S c G, G open, I G[ < oc, > O. If v ~ S we can choose an open cubic interval centered at v so that

p(g(Q(>'))) ~2(:') ~ o, I O(:')l < ~

By the lemma we can choose a sequence {Q~.} from among such sets (Q(y))~,

satisfying

S c U Q, = 6. ~:ze. < Oln). P(g(Q~)) , - I 0 , I

. . . . . < / - ;

Therefore

p(yls) <_ f, (y(U Q,)) < p(U y(Q,)) k

< ~o(,,) I U Q~ I < ~ot,,) I~1. k

Since e is arbitrary, p(g(S)) = 0 and the theorem is proved.

REMARK 1. It is rather easy to show, by means of the mean value inequality

that if in the theorem X = II~", p is the exterior measure associated to Le-

besgue measure and y e ~ 1 then

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.--~- = ~.o/~v ~ = 0

implies that Y0 is a cri t ical poin t in the sense of the above definit ion. So the

theorem of Sard is an easy consequence of the theorem.

REMARK 2. It is easy to show that if X = Nt", p is the exterior measure and

9: 1/ --, X is one- to -one and t ransforms Lebesgue measurab le sets into Le-

besgue measu rab l e sets, then the limit

lim I y(Q(:.'))I

exists for a lmos t every y ~ I/.--Here Q l y ) ~ v means that one considers all

open cubic intervals centered at v and makes their d iameters shrink to O.

In fact, for y > s > 0 and B a bounded subset ot [R" we cons ider

Ar, . . . . = { r e V c~ B : 3Q~.(.~') --' v, ~Oth ' ) ~ r, 10(Qk(y))il OAv)[ > r > s > [ q~Q~'I.~))I Q~'~Y) I I'(j.

It I Pi~ means the exter ior measure of the set P, we have i A~.,],, = 0. Let

[A~,I,, = a. We take an open set G, G c A,.,, I GI < ~ + r/, for 17 > 0 given. �9 . {Q~(~)~ with F o r v eA~, we have a sequence * '

Q*(~,)--, v, O~ty) = 6, ly(Q~'/y))I �9 " I O~'~y) l < ~"

We apply the theorem of Vitali and so we obta in a disjoint sequence (Qt,, (y))

from a m o n g (Q'~(v))~. ..,~,4,. such that

.~t~+ ,7) > ~IGI >- sZIQ~'.I--- Y~I~IQ~')I k' / ,

and

IZ,.,- U QT = 0 . ,~

Let now v*~ ( ~ Q~',)c~ A,.~ = C. Observe that almost every point in A,.~ is ,~"

also in C. F o r v* we have a sequence {Q~(y*)] with Q~(.v*) ---, 3'*, Q~(.v*) c

Q~*, E (Q*,(y)) for some h such that

I a(Qk(v*))l I (2kt.~'*)l > r.

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We apply again the theorem of Vitali to ob ta in {Qk.} ano ther dis joint s equence

from a m o n g (Q.(Y*).)/~c,k such that we can write

~=lA,~l~=lcle < _ l c - a Q k + c~(UQ~) e r

=[cn(UOk) _< ~lQkl < . LY'lgt0k)lr ~ e

1 S -< --7_ ig/Q~')i-< --Ca + n). r h r

rls ~4ence cc <

r - s Since r/ is a rb i t ra ry , ~ = 0 and so I Ar~ Iv = 0.

REFERENCE

[ l ] M. DE GUZMAN, A covering lemma with applications to the dijJerentiability of measures and singular integral operators; Stud ia Math. 34 (1970), 299-317.

Pontificia Universidade Cat61ica Rio de Janeiro

Facultad de Ciencias Maternaticas Madrid

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