PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 48, Number 2, April 1975
A NOTE ON A COROLLARY OF SARD'S THEOREM
JOHN C. WELLS
ABSTRACT.
Corollary.
on a compact
Lebesgue
subset
measure
The
and
of this
Let
m(K)
is infinite].
note
continuous
function.
X
measure
Then
a Borel
there
f\K,
Lemma
subset
sets,
a similar
version
the
of this
X a measurable
of X, less
theo-
than
oo. Let
subset
of
C = \y\f ~ (y)
two lemmas.
X a compact
exists
subset
X' C X with
the zth coordinate
of Rn and
/(X')
= /(X),
f a
/) „,
072t?
exists
y £ X with
of x. Define
y, < X. — 1//
for /' = 1, 2, 3,•• •,
and
fiy)
= fix)
or
y < x. - 1//' and fiy) = fix) or • • • or y £ X with y.
, = x _., y <x
is not contained
that
will require
- i//
each j, X = X - (J. A . is Borel.
Clearly
Then
set.
x. denote
y £ X with y, = x,,
y1 = InfU1|/U)
is infinite].
f Lipschitz,
f : X —» Rm with
A . = \x\ x £ X and there
= x., • • •, y
(y)
272(C)= 0.
to one and
Let
C = !y|/—
f : K —> Rn with
1. Let
Proof.
Let
is to show
The proof of the theorem
Lemma
is the following:
functions.
the Lebesgue
Then
theorem
be a smooth (i.e. f e C , k > 1) map defined
K of R".
for Lipschitz
Theorem.
of Sard's
of C is zero.
purpose
rem holds
R"
A corollary
Let f: X -» R
= /(x)|, •••,yn
in A . for any
and fiy)
= fix)].
Since
If x e /(X) observe
= Inf[zJZl=y1,...,^_1
/ and also
fiy)
A . is compact
for
that y defined by
= yn_1,/U) = /(x)|
= fix).
Thus
/(X')
= /(X).
is one to one.
2. Let
f: X —> Rm be a continuous
of R". Then there exist
map from
X a measurable
X', X" with K" C X' C X, X" and X' Borel
f\K„ one to one, m(K - X') = 0 and f(K") = /(X').
Proof.
we can write
Received
There exists
K = (J
an FΠset
K
by the editors
where
X
X', with
is compact
X' C X and 772(X- X') = 0, and
for each
April 3, 1974 and, in revised
22. By Lemma
1
form, June 26, 1974.
AMS(MOS)subject classifications (1970). Primary 26A63, 28A75.
Copyright © 1975. American Mathematical Society
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513
514
j. c. WELLS
there
are Borel
each
sets
X'22 C X 22 with
72. Define recursively
/(X') 22 = 'f{K 22) and
f\„,
' 7\
'
'
one to one for
22
D x = K\, ■• • , D k = Dk_x U {K'k - f ~ lifiDk_)))).
Then X" = Uz. Dk IS Bore^ fiK") = /(X') and f\K„ is one to one.
Proof of Theorem. Suppose 272(C)= a > 0. Let A = / _ (C) and find by
Lemma 2 zV'¡ CNjCA,
with /V" and N\ Borel,
tz2(Aj - N\) = 0, /|N«
one
to one and f(N'\) = fiN\). Since / is Lipschitz,
m{f{Al - N\)) = 0. Therefore 77z(/(zV'j))= a. Now /(Aj - zV'J)= C by the definition of C. Thus for
£ = 2, 3,'
• • by letting
A, = A, _ 1 — zV',_ j we can repeat
to find N"k C N'kCAk
Suppose
that
with /|N»
N,
Remarks.
O N, = 0
to a requirement
coordinate
of a continuous
a space-filling
On the other
lowing
argument
Let
For each
x e A
- /(*)|
for /;
k = k , this
that
Then
for
/-
Suppose
A;
are arbitrarily
< \y - x\ when
with
there is a disjoint
'
collection
M/(A2))<m/7/A2-U
over
/ ~ (y) must consist
a.c.
zero.
intervals
Thus
/
y £ I.
suffice
as the fol-
function
on /. Then
m{f{A ,)) = 0 since
positive
containing
is a Vitale
number.
x such
the collection
covering
that
J
of A
of
so
1/n I C A' with 2tz(A^
-II W22 / 22) = 0. But then
2
lnX\ +rn(f(\J I \\ < 0 + 2r, ■£ length(M < 2n.
Since T) is arbitrary, m{f{A2)) = 0. Finally,
is infinite!
will
ij be an arbitrary
A.
be
/ be the first
for each
valued
y is in the intervals.
x varying
let
cannot
/ onto the unit square
of measure
small
for each
in the theorem
continuity
/ is a real
on a set
and w(/(/V'¿)> = a.
?tz(X) = <x>la contradiction.
(y) is infinite
72 = 1 absolute
argument
m(N"A > a/L
To show this
A., = {x| / (x) = 0i and let
intervals
implies
/ be Lipschitz
of continuity.
except
there
then
map of the unit interval
shows.
/ is a.c.
|/(y)
for
curve).
hand,
/ is differentiable
all such
constant
The requirement
weakened
(i.e.
one to one, fiN'j) = /(/V¡)
L is the Lipschitz
X. But since
the above
of isolated
points
if y e /(/) - {{A J - fiA^
and hence
be finite.
Thus
then
7?2Íy|/_
(y)
= 0.
REFERENCES
1. John
Milnor,
ginia, Charlottesville,
2.
A. Sard,
Topology
from
Va., 1965.
The measure
Math. Soc. 48 (1942), 883-890.
the differential
viewpoint,
Univ.
Press
of Vir-
MR 37 #2239.
of the critical
values
of differential
maps,
Bull.
Amer.
MR 4, 153.
DEPARTMENT OF MATHEMATICS, CALIFORNIA STATE UNIVERSITY, NORTHRIDGE,
CALIFORNIA 91324
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