THE GAUSS-GREEN THEOREM · THE GAUSS-GREEN THEOREM BY HERBERT FEDERER 1. Introduction. All...

THE GAUSS-GREEN THEOREM

BY

HERBERT FEDERER

1. Introduction. All conventions of our papers on Surface areai1) are again

in force. The positive integers m = n which were fixed throughout SA II are

now so specialized that m=n — 1, «2:2. The corresponding(2) function $ is

an (re —l)-dimensional measure over Euclidean «-space, which reduces to

Carathéodory linear measure if « = 2.

The starting point of the present article is the definition of the exterior

normal of a subset 4 of «-space at a point x in «-space. A glance at 3.1-3.4

below will convince the reader that the existence or nonexistence of this

vector is a local, geometric property of the set 4 at the point x. None but the

most elementary topology enters into this definition in which the boundary

of 4 is never even mentioned. As will be reaffirmed in 3.6, v(A, x) is the ex-

terior normal of 4 at x whenever this exists, otherwise v(A, x) =6, the zero

vector.

With these definitions of surface measure and exterior normal at hand, we

are led to investigate the validity of the Gauss-Green formula

(1) f Djf(x)dx = Ç f(x)Vj(A, x)d$x.

Here 4 is an open subset of En,j is a positive integer between 1 and «, v¡(A, x)

is the jth component of v(A, x), and DJ is the partial derivative of/ in the

direction of the jth unit vector. In this connection we shall always make the

assumptions that both integrals are finite, that the boundary of 4 has finite

4> measure, and that/ is absolutely continuous within the closure of 4 along

almost all lines in the direction of the/th unit vector(3).

It is seen from Theorem 6.4 that (1) is true if the above conditions are

satisfied and if, in addition, the boundary of 4 has a certain(4) type of regu-

larity. However we are able to prove much more in the special case w = 2. In

fact it is shown in §7 that the boundary of every open set in the plane has

the required regularity if it is of finite Carathéodory linear measure. Thus

(1) holds in the plane under the conditions of the preceding paragraph. The

Presented to the Society, April 29, 1944; received by the editors March 28, 1944.

(») See §2 of Surface area. I, Trans. Amer. Math. Soc. vol. 55 (1944) pp. 420-437, and

§§2, 3 of Surface area. II, Trans. Amer. Math. Soc. vol. 55 (1944) pp. 438-456. We hereafter

refer to these papers as SA I and SA II.

(s) See SA II, 2.1. The special case » = 3 is also treated in SA I, 3.1-3.3.

(3) See 6.1 and 6.3.

(*) See 3.8-3.14 and the condition (II) of 6.4.

44License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

THE GAUSS-GREEN THEOREM 45

question whether this much is true in higher dimensions is left unanswered.

The above mentioned regularity of the boundary of A is called i> restrict-

edness. Its local character is clear from Definition 3.12. Anthony P. Morse

and John F. Randolph have recently(5) introduced and investigated this con-

cept in the case of plane sets. The author wishes to express his sincere thanks

to them for the opportunity to read their paper in manuscript. He has freely

used their methods and results. Their paper and Randolph's thesis(") have

been his main source of interest in the subject matter of the present article.

§2 contains a theorem about the transformation of integrals. A generaliza-

tion of the strong form of Cauchy's Theorem, concerning the integral of an

analytic function "around" an open set, is proved in §8.

2. Transformation of integrals.

2.1 Theorem. If(I) 

measurable sets of finite <b measure ;

(II) ^ is such a measure over the metric space B that closed subsets of B are

\¡/ measurable and every \p measurable set is contained in a Borel set of equal

ip measure;

(III) gis such a function on A to B that

E [g(x) £ Fj

is <f> measurable for every closed set F £73 ;

(IV) u is such a x

for every <f> measurable set XCA ;

then

ff(y)N(g, X, y)#y - j f[g(x)]u(x)d<t>x

whenever X is a<p measurable subset of A and f is such a \p measurable function

that — oo ̂ /(y) ^ » for \j/ almost all yinB.

Proof. We fix a rectifiable subsets of the plane, Trans. Amer.

Math. Soc. vol. 55 (1944) pp. 236-305. We hereafter refer to this paper as RM. The bibliography

of this paper gives references to previous articles on this subject, in particular the work of A. S.

Besicovitch on plane sets of finite Carathéodory linear measure.

(') John F. Randolph, Carathéodory linear measure and a generalization of the Gauss-Green

Lemma, Trans. Amer. Math. Soc. vol. 38 (1935) pp. 531-548. The first attempt in this direction

was made by J. Schauder, The theory of surface measure, Fund. Math. vol. 8 (1926) pp. 1-48.

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

46 HERBERT FEDERER [July

P = C E [0 < u(x) < oo 1.X

Whenever r and / are functions on 4 and B respectively we denote

a(r) = I r(x)u(x)d<t>x,J p

ßif) = j f(y)N(g, P, y)diy.

Let £ be the set of all such \p measurable functions/ that — w g/(y) ^ oo for \p

almost all y in B, and take

H = FE [/(y) = 0 for y G B - g*(P)].

We further define for each/G £ the function /on 4 by the relation

f(x)=f[g(x)] for x E A,

and divide the remainder of the proof into four parts.

Part 1. g*(P) is expressible as a countable sum of \p measurable sets of

finite yp measure.

Proof. Use (I) to select -i

Pk = PE [u(x) < k] for k - 1, 2, 3, • • •

and¿-i ¿-i

g*(4/i>*) - £ [N(g, P/4», y) £ 1],

» > B-^yP*) 2: f u(x)d<t>x = f iV(g, £,-4,, y)d*y >: ̂ [g*(43£t)]

for every pair of positive integers / and k.

Part 2. The setPE[g(x)EY]

X

is <p measurable for every \p measurable subset Y of g*(P).

Proof. Let G be the family of all those subsets F of £ for which the set

P E [gix) E Y]X

is <p measurable.

Evidently G is closed to countable addition and to complementation.

1945] THE GAUSS-GREEN THEOREM 47

Hence (III) implies that every Borel set of 73 is a member of G. Next we infer

from Part 1 and (II) that every \p measurable subset of g*(P) is expressible(7)

as an £„ plus a set of \p measure zero. We shall complete the proof by showing

that every subset of g*(P) of \p measure zero is a member of G.

Suppose YCg*(P), yp(Y) = 0 and

X = P E [g(x) £ Y].X

Use (II) to select a Borel set Yx of 73 for which F£ Yx and ^(Fi) =0. From

the preceding paragraph we see that Fi£G. Let

Xx = PE[g(x) CYx]

and use (IV) together with the relation ip[g*(Xx) ] á^( Fi) = 0 to infer

0 = J N(g, Xx, yWy = J u(x)d<px.

But XxCP so that <b(Xx) =0. Since XCXU we conclude <p(X) =0 and F£G,

Part 3. fCH implies ß(f)=a(f).Proof. In case / is the characteristic function of a \p measurable set

F£g*(£),weletX = £ [g(x) £ F],

X

note that/is the characteristic function of X, and use Part 2 and (IV) to infer

«(/) = f «(*)d** = f tf(g, £*, y)#y = f arfo P, y)¿*y - ß(f).J PX J J y

But the functions a, ß and ~ are additive, homogeneous, and continuous

with respect to monotone convergence of nonnegative members of their do-

mains. Hence

ß(f) = a(f) for every nonnegative / £ H.

Now suppose / is an arbitrary member of 77. Let /i and f2 be such non-

negative functions in H that f=fi—f2. Then /i and f2 are nonnegative func-

tions for which f=]i—]i, and we infer from the definition of the Lebesgue

integral that

ß(f) - ß(f>) - ß(fu = *(Jù - «(70 = «(/)•Part 4. If/££, then ff(y)N(g, C, y)dnpy =jc](x)u(x)d4>x.Proof. Select /z£77 so that

Ky)=f(y) for y£g*(£). *(y) = o for y e b - g*(P).

C) See Theorem 3.13 of RM.

Check the relations:

(i) ßih) = «(*);

h(x) = f(x) for x G P,

(2) «(A) = «(/) = f Jix)uix)d4»x;J c

0=| u(x)d$x = f 2V(f, C - P, y)#y,»J c—p J

N(g, C, y)=N(g, P, y) for \p almost all y in £,

(3) j f(y)N(g, C, y)d^y = ß(f) = /5(A).

Combine (3), (1), and (2).

3. Definitions.

3.1 Notation.

n

«•y=E*/:y/ for xEEn,yEEn,

KTX= EnE[\z- x\ <r] for a; G En, r > 0.

3.2 Definition. We say u points into S at x if and only if SC£n, xEEn,

«G£n, |«| =1, andll

|h,-s|lim —¡-¡— = 0,r-0+ | ET |

where, for r>0, Hr is the hemisphere

£l£ [(z - x).u > 0].

3.3 Definition. We call « an exterior normal of S at x if and only if u

points into (£„ — S) at x and ( — u) points into 5 at x.

3.4 Theorem. If u and v are exterior normals of S at x, then u=v.

Proof. For wEE„ and r>0 we write

H? = KTX E[(z- x).w> 0].

First we prove

(i) hu3h7 = 0.

In fact the denial of (1) implies

| 77r77r | = r \ExHx | > 0 for r > 0,

and since

\HrS\ \Hr -S\lim —¡——r— = 0, lim -¡- = 0,

r^0+ \H«\ r-K)+ \H-V\

it is obvious that1U —V I —VU\

HrHr S\ ,. \HT Hr ~S\lim -j-¡— = 0, lim —¡-¡-= 0.

r-o+ £"£•-" ,-o+ H-VHU

Adding we obtain

hm i-r = 0r-o+ H"H-

r r

which is false. Thus (1) is proved.

Now let z=x+u— v. Then the assumption icKl implies (z — x)-u

= (u—v)'U = l—vu>0, zCH3; thus (1) yields the inequality 0^,(z—x)> ( — v)

= (u — v)' ( — v) = l—U'V>0, which is false. Hence wv^ 1 and u=v, because

3.5 Notation. We fix the points 6 and i so that

6 = (0, • • • , 0) £ £„, i = (0, • • • , 0, 1) £ £„.

3.6 Notation. For 5C£n and *££„ we define v(S, x) as follows:

If 5 has a unique exterior normal u at x, then v(S, x)=u; otherwise

v(S,x)=6.3.7 Notation^).

{x} = E[y= x].

3.8 Definition. If SCEn and #££„, then sgn (S, x) is the closure of the

set

vGt KI y - *|)where T = S-\x}.

3.9 Definition. If 0 is a measure over £„, 5££„ and *££„, then we

define the upper and lower <¡> density of S at x by the relations

a . <t>(SKrx) v <t>(SKrx)D$ (S, x) = lim sup-) Dj, (S, x) = lim inf-•

r-o+ y(KT) r-0+ T(£r)

From Definition 2.1 of SA II it is seen that y(KTx) equals the (« —1)-

dimensional Lebesgue measure of an (« —l)-dimensional sphere of radius r.

(8) Throughout this paper we use braces only in this sense.

3.10 Definition. If <j>'is a measure over £n, 5C£n and xEEn, then

dir* (S, x) = n sgn (5 - ß, x)ß&F

where

£ = E [Dt(ß, x) = 0].

3.11 Definition. We say 5 is restricted at x if and only if

Dt(S, x) > 0

and there is a point zEEn for which \z\ =1 and

({z} + {-z})dir,(S,x) =0.

3.12 Definition. We say z is 0; zEEn, \ z\ = 1;

dir* (5, s) C£ b'Z = 0].

3.13 Definition. x' = (xi, x2, ■ • ■ , xn-i)EEn-i for xEEn,

S' = E {*'} for SC£n;

(y o t) = (yx, y2, • • • , yn-i, t) E £» for y G £n-l, t G £i.

For 4 C£» and k = l, 2, 3, • • • we let

AÎ(4) = En £ [0 < * - xn < k'1 implies (*' o t) G 4],

a!(4) = £„ £ [0 < Xn - t < k'1 implies (x' o f) E A],X

A+(4) = ¿ AÎ(4), A"(4) = ¿ Al(4).fo-l i:«.l

3.14 Definition. For each function g on £„_i to £„ we define the function

rg on the setE [Jg(y) > 0]

to £„ as follows:Suppose ./g(y) >0 and L is the approximate differential of g at y. For each

integer & between 1 and k, strike out the Bth row of L to obtain the minor M*.

Let Tg(y) be the point of £„ whose «th coordinate is

k(y)]* = (- l)n+*(det Mk)/Jg(y).

It is well known that rg(y) is perpendicular to the columns L1, L2, • • • ,

Í,»-1 of I.

3.15 Remark. If .4 ££„ with $(A) < », then

D<¡>(A, ï) ^ 1 for $ almost all x in A.

If furthermore B is a Borel set contained in A, then

DÎ(A, x) = £>f(73, x) and dI(A, x) = I>J(73, x)

for <ï> almost all xin B, and

dir$ (73, x) = dir* (A, x)

for <ï> almost all xin B.

These statements are the analogues of 5.16, 5.15 and 6.11 of RM. Their

proofs are likewise analogous: $> replaces L, and y(KTx) replaces 2r.

Two sets, one of which is a measure hull of the other, have everywhere

the same densities and the same dir. Hence the assumption that £ is a Borel

set can be omitted.

4. The exterior normal.

4.1 Lemma. If S is an open(*) subset of £„, | S| < », | S'| < », and t<\,

then there is a number u such that u<l and

\x\^u\s\ implies \X'\^t\S'\

whenever XCS and X is of class F*.

Proof. We assume |S'| >0 and define

h(y) = EiE [(y o v) G S] | for y £ S',

A(v) = S' E [k(y) > v] for - » ^ v g «,

w = sup£ [|4(î;)| è f I S'IL

B = S' E [h(y) ^ w].

Clearly A(w)£73. Furthermore \S'\ < oo implies

\A(w)\ = sup \A(v)\ £ t\S'\ ^ inf |4(»)| = \b\.

Hence we can and do select a Lebesgue measurable set T for which

A(w) CTCB and | T\ = /|S'|.

Next we takefrh(y)dy

u = —j—j— •_ \S\

(*) It is sufficient to assume that h(y) >0 for almost all y in S', where h is defined as in the

proof. This condition is really necessary in case 5 is measurable. In case 5 is nonmeasurable the

lemma is obvious.

In order to prove that

(1) « < 1,

we suppose u—1. Remember that S is open and h(y)>0 for yES'. Check

the relations:

f h(y)dy as | S\ - f A(y)dy, r C 5'; f A(y)dy = 0;J T J S' J S'—T

\S'-T\=0; t\S'\=\T\=\S'\, 0<|5'|<oo; j = 1.

Since t<l, we have verified (1).

Now choose XC.S so that X is a set of class £„ for which \x\ =u| S\ and

suppose |X'| « | 5'|. Then

f A(y)¿y + f A(y)áy = f A(y)dy = | X | = « | S \J X'-T J X'T J X1

= f Hy)dy = f A(y)dy + f A(y)dy,•/r J T—X' J TX'

V>\X'-T\t\ h(y)dy^\ h(y)dy 2: w| T - X'\;J X'-T J T-X'

| X' - T | + | X'T | = | X' | < 11 y I = I TI = I T - X' | + | TX' \ ;| x' - T | < | T - X' | and w | Z' - T \ = w | T - X' | ;

w = 0, 4(w)=5'; l3"| -|4(w)| Sl|S*|;

t= 1.

But ¿<1, so that |X'| ^t\S'\.

4.2 Theorem. //£ is /Ae boundary(10) of the open(u) setACEn, v(A,x)^d

and D$(B, x)Sl, then Dj(B, x) = l and v(A, x) is 4> perpendicular to B at x.

Proof. Since all the notions involved are invariant under distance preserv-

ing transformations of £„, we may assume x = 0 and v(A, x)=i. It will be

sufficient to show that D$(B, 6) = 1 and

dir* (B, 6) C £ [ | yn | < e] for every e > 0.

Let e > 0 be given and take

ß = BE [\zn\ <*\z\].

Since D$(B, 6) = 1, we shall complete the proof by showing thatD$(ß, 6) = 1.

(10) We say T is the boundary of 5 if and only if T= [closure S] ■ [closure (En — S) ].

(u) The hypothesis that A be open is unnecessary. This may be seen by applying the theo-

rem to the open set .4x = Int A = (A +B) —B, whose boundary is a subset of B, and for which

J»(4i, 3c)=v(j4, x).

For this purpose choose a number t such that 0<<<1. For each r>0 let

SÏ = Krx E [0 < z„ < « | z | ], S7 = £* £ [0 < - z„ < « I z I ],

Cr = (sty = (57)'

and use lemma 4.1 to select a number « such that 0<w<l and \X'\ ^t\Ci\

whenever X is either a subset of Sx+ of class £„ with | X \ ^ « | Sf \, or a sub-

set of Sr of class Fc with | X \ ^ tt | Sr |.Use 3.6 to choose p>0 so that 0<r<p implies

\S+r - A\ = u\st\ and | S~A | £ u\S~\.

Now fix such a number r and denote

X+ = £ [rz £ (S* - 4)], Z~ = £ [rz £ S7¿].

We see that

\X+\ =r-»|S^-4| è «r-\st\ =«|st|;

hence

\{St-A)'\ =r»-1\(X+)'\ fc*r-»|Ci| =t\Cr\;

similarly | (Sr4)'| ^/| Cr\, and consequently

| (S+r - A)'(Sr~A)' | = | (St - 4)' | + | (S7A)' I - | (St - A)' + (STA)' |

5; 2t\ CT\ -\Cr\ = (2t- 1)| C,|.

But (Sf— A)'(S7A)'C(ßKTx)', because every line segment which joins a

point of A to a point of £„—.4 must have a point in common with B. Thus

HßKrx) ä I (ßKx)' I è (2* - 1) I C,| - (2t - l)y(KTx).

Since r was an arbitrary positive number less than p, we infer that

D%(ß, x) = 2t- 1.

Let/->1.

4.3 Theorem. If X is a Lebesgue measurable subset of £„, F££„_i, and g

is such a Lipschitzian function on £„_i to £„ that

g*(Y) C A+(X)and

[g(y) ]' - y for y £ JSn-i,

then rg(y) points into X at g(y) for almost all y in F.-

Proof. Let

Tk = [\t(X)]'-Y for ft- 1,2,3,-..

and check that

Y = [g*(Y)]'Y C [\+(X)]'Y = ¿ Tk.k-l

Since g is differentiable almost everywhere in E„-i, and Tk has density 1 at

almost all of its points, we shall complete the proof by verifying the following.

Statement. If A is a positive integer, yETk, g is differentiable at y, and Tk

has density 1 at y, then rg(y) points into X at g(y).

Proof. Let L be the differential of g at y and denote c=rg(y). Obviously

cnJg(y) = 1. Let x =g(y) and

HT = Kx E[(z- x).c> 0] for r > 0.

Choose 17 >0.

Select e>0 so that ¡£fi— Z7X| èv\Hi\, where

Ur = £x £ [(z - x) -c > « I 2 - x I ] for r > 0,

and then take p>0 so that ftp(l + É+||l,||)<l and

I g(w) — g(y) — L(w — y) I = € I w — y I whenever | w — y | < p.

Now suppose 0<r<p.

We first prove:

(1) IfwG£*and (w o t) G UT, then (moíjGI

In fact the hypotheses of (1) imply that \w—y\ ú\z — x\ <r, where

z=(w o t), and that

ft-» >(1 + a + |fx||)f 2: I c - x\ + \ g(y) - g(w) \ = \ z - g(w) | ¡S [f - g(w)].

= Jg(y)[z - «(»)]•« = -fg(y)[(z - *)«c- [g(tp) - g(y)].c]

> /g(y)[e| z — *| +1 c-L(w - y) \ — e| w — y\]

= Jg(y) [|z— ac|—|w— y|]e=0.

Since zn = t, this implies k~l>t— [g(w)]n>0. But g(w)EA-it(X), so that

(wot)EX. Thus (1) is proved.

Next we define the set W as follows:

w G W if and only if »G Ui and

I £x £ [(w o /) G £7, - X] I = 0.

From the Fubini Theorem we know that W is Lebesgue measurable, and

19451 THE GAUSS-GREEN THEOREM 55

from (1) we infer UiTkCW. Hence

| VI - W\ - | VI | - | W\ á I VI I - I U'rTk\,

and we apply Fubini's Theorem again to obtain

| Ur - X\ = (diam Ur)- | Ul -W\û 2r( | Ui | -| £7rT»| ).

Consequently

|77r-X| <|g,- C7,| l^r-A-l

£r| - f|7Ji| r"|Hi|

\Ei-Vi\ , 2r| W| | £7,'| -I I7r'r»|

. TJi | r« I Hx | | £7,'

Hr-X\r-x\ , 21 vi| / |t/r'n|\

Remembering that r was an arbitrary positive number less than p, and

that Tk has density 1 at y, we conclude

r \E'~X\ <lira sup-¡——:-s n,

r-0+ | 7ir |

Since r/>0 was freely chosen, the last relation implies

| Hr - X |lim -:-j-= 0,r—0+ I 77r I

so that c points into X at x.

4.4 Theorem. If X is a Lebesgue measurable subset of En, F££n_i, and g

is such a Lipschitzian function on £„_i to E„ that

g*(Y) C A-pQ

and[g(y)Y = y f°r y G &n~u.

then —rg(y) points into X at g(y) for almost all y in Y.

4.5 Theorem. If A ££„ and f is the function on £„ to £„ such that f(x)

= v(A, x) for xCEn, then f is a Borel measurable function(12).

Proof. LetS = £„£ [|«| = 1].

(12) By this we mean that the counter-image of every closed set is a Borel set. This condition

is satisfied if and only if each of the coordinate functions /i, /2, • • • , /„, which are defined by

the relation/(*) = (fi(x),fi(x), • ■ • ,fn(x)), is a Borel measurable function in the classical sense.

Since the range of / is a subset of S+ {d}, it is sufficient to prove:

If C is a closed subset of 5 and

G = £ [f(x) E C],x

then G is a Borel set.

Suppose C and G are so fixed and denote

R = EnXS = E [x G En and u E S],

Hrix, u) = Kx E [(z - x) • w > 0] for r > 0, (*, m) G £.

For each r>0 let g, be the function on R such that

| Hrix, u)A\+\ Hrix, - u) - A I&■(*, «) =-r—i- for (x, u) E R.

I Âi I

We denote the symmetric difference of two sets X and Y by

(X, Y)= (X-Y) + (Y- X)

and infer that

| gT(x, u) - gr(y, v) |

(1) | <gr(*. «), #r(y, ¡0)1+1 <g,(», - «), gr(y, - i>)) 1

k:

for r>0, (*, m)G£, (y, »)GJ?.For each positive integer j let

and let

T¡ = E E £n £ [0 < r < Ä-1 implies gr(*, m) g /-»],«6c 4-1 x

t = n ry.y=i

The remainder of the proof is divided into two parts.

Part 1. If j is a positive integer, then T¡ is of class £„.

Proof. It follows from'(l) that gT is continuous for each r >0. Consequently

the set

<2*. n *£>(*,«) = ]-']

is closed for each positive integer k. Likewise closed is

H = R E [wGC],

and, letting P be the function on R to £„ such that

P(x, u) = x for (x, u) £ R,

we easily see that

T, = ¿ P*(HQh).*—1

But P projects sets of class Fc into sets of class £„.

Part 2. G = T.

Proof. Clearly

G = £„ £ [ lim gr(x, u) = 0 for some « £ C],

T = £„ £ [ inf (lim sup gr(x, u)) = 0].* u£c r->0+

Now fix a point #££„ and let

Ar(«) = gr(x, u), h(u) = lim sup h,(u) for « £ S.r->0+

From (1) we infer that

| Är(«) - Är(») J ̂ -j—-j-■-—-—

whenever m£S, vCS. Thus the functions A, are equicontinuous. Hence h is

continuous and assumes its minimum on the compact set C. Consequently

T = £„ £ [lim sup gr(¡c, w) = 0 for some u £ C],x Í--0+

which implies T = G.5. Restrictedness. The following two theorems are analogous to some re-

sults of §7 of RM. Since the proof of our theorems is an almost literal repeti-

tion of arguments used by Morse and Randolph, we merely illustrate what

sort of changes have to be made:

Replace L, <p, 0° by "i, and judiciously replace 2r by y(K'x). Use the rela-

tion x = (x' o xn) for *££„, instead of the relation x = (*i, x2) for xCE2, for in-

stance

(x, a, X) = £ [ | y' — x' \ < a and | yn — xn \ < Xa],

I x | = | x' | + | xn |, (x - xj)j = (O,-

for #££„ and j = l, 2, • • • , « — 1.

5.1 Theorem. 7/S££„, $(S)< °o and

Di(S, x)>0, ({i} + { - i}) dir* (S, x) = 0

for <ï> almost all x in S, then there are Borel sets Tx, T2, T3, ■ ■■ ££n-i and

Lipschitzian functions gx, g2, g3, ■ • ■ on £n-i to En such that

*[s- Ê*î(r*)]-o.

[gk(y)Y = y /or yG£n-i, ft-1,2,3,-...

5.2 Theorem. 7/ 5C£n, $(S) < » awd S is $ restricted at $ almost all

of its points, then there are Borel sets Ti, T2, T%, - - • C£B-i and Lipschitzian

functions gi, g2, g3, ■ ■ ■ on £„_i ¿o £n smcA /Aa£

*[s- ¿«î(r*)] = o.

5.3 Theorem. If <$(£) < », T is a Borel set of En-i and f is such a Lip-

schitzian function on £„_i to En that f*(T)Ç_S, then there is such a Borel set

ACT that

*[f(T) - f(A)] = 0,

and rf(y) is <3? perpendicular to S at f(y) whenever yEA.

Proof. Let

4i- TE[DÈ(S,f(y)) = l],

42 = 4i£ [/is differentiable at y with Jf(y) > 0],

A3 = 42 £ [// is approximately continuous at y],

and note that Ai, A¡, A3 are measurable sets. Choose a Borel set 44C43 for

which |43— 44| =0, and use 3.15 of this paper and 4.5 of SA II to check

*[/*(r) - f(Ai)] = 0, *[/*(4i - 42)] = 0,

|42-44| = 0= $[/*(42-44)], *\j*(T) -f*(AJ] = 0.

Next select(13) a Borel set 46C44 such that /*(46) =/*(44) and / is uni-

valent on At. Finally let A be the set of those points of At at which At has

Lebesgue density 1. We see that A ET, A is a Borel set and

|46-4| =0= $[/*(46-44)], *[/*(D -/*(4)] = 0.

Now pick a point yG4. Abbreviate x=f(y), c = rf(y). In order to show

that c is $ perpendicular to 5 at x, we proceed as in the proof of 4.2: Let

e>0 be given, take

ß-SE[\ (z -*).<;[ á c| i- *|],

and complete the proof by verifying (1) below.

For this purpose choose a number / so that 0</<l.

(ls) See H. Fédérer and A. P. Morse, Some properties of measurable functions, Bull. Amer.

Math. Soc. vol. 49 (1943) p. 276, Theorem 5.1.

Let L be the differential of / at y, let M be the inverse of L, and choose

a number r¡ so that

0 < v = «(IMI-1 - v) and (1 + tjIImII)"-1/ é 1.

Abbreviate s=(l+i7||M||)-1 and select 5>0 so that \f(w)—f(y)—L(w—y)\

^7¡\w—y\ whenever \w—y\ ^5.

We define

Ur = £„_i E[x + L(w - y) £ Kx] for r > 0,

and note that diam 7Jrg2||lf||r.

NowsupposeO<2||M||rg5andw£^t/r„s=/(w).Then |w-y| ^2||ilf||rs

^ ôs<ô and |£(w—y)| iSrs. Hence

\z — x\ £rs + rj\w — y\ g rs + rj| M[L(w — y)]\ á rs(l + i?||Ai||) = r

and zC-.K'x- Furthermore

| z — x | ^ | L(w — y) | — 171 zc — -y I è ||-^||_1 \w— y\ — i\\w— y\

= (\\M\\-1-v)\w-y\,

I c-(z — x) I = I C'L(w — y) + c[z — x — L(w — y)] |

= I c« [z — x — L(w — y)] I ̂ I z — x — L(w — y) |

úv\w-y\ afdWI"1-»)-»!■- «I Si «I*- «I.which implies z£/3. Accordingly z£/3£^.

We have proved that

/*(4L7„) C ßKx for 0 < r < 5(2||m||)_1.

For all such r we have

$0370 £ *[/*(il£7„)] = f N(f,AUr., z)d$>z = f Jf(w)dw,

y(KTx) =\Ur\lf(y) = s^\Ur.\jf(y),

HßKTx) n_i |4t/„| fAuJf(w)dw _

y(£p =5" I C7„| 7/(y)|¿17„| *

Since A has density 1 at y and Jf is approximately continuous at y we con-

clude

DÏ(fi, x) à s»-1 = (i + ^IImII)1-" = t.

From the arbitrary nature of / we finally infer

The proof is complete.

(1) D%(ß, x) ^ 1

5.4 Remark. If y is any point of the set 4 of Theorem 5.3, then

dir* (S, f(y)) = £„ £ [| u\ = 1 and M-r/(y) = 0].

This fact will not be used in this paper.

5.5 Theorem. If ZQEn, $(Z)< =° and Z is $ restricted at <£ almost all

of its points with

({*} + {-•}) dir» (Z, *) s*0 for xEZ,then | Z' | =0.

Proof. In view of 5.2 we assume that we are dealing with the special case

in which Z=f*(T), where £ is a Borel set of £„_i and/ is a Lipschitzian func-

tion on £„-i to £„. We select 4 in accordance with 5.3, here S = Z, and define

the function g on £„_i to £„_i by the relation

g(y) = [f(y)Y for yG£n-i.

Evidently g is Lipschitzian and

\Z'\ á | g*(A) | + 1>[Z - /*(4)] = J JV(«, 4, w)dw = j Jg(y)dy.

We shall complete the proof by showing that Jg(y) =0 for every yEA.

Pick yG4 and let L be the approximate differential of / at y. Since

rf(y)EEn we know that Jf(y)>0 and the points L1, L2, ■ ■ ■ , Ln~x are lin-

early independent. Using 5.3, our present hypotheses, and 3.15, we see that

rf(y) • i = 0 and rf(y) •£' = 0

for .7 = 1, 2, ■ • • , « —1. Hence the points i, L1, L2, ■ ■ ■ , £n_1 are linearly

dependent. Since the last re —1 are independent, we can find numbers

t\, h, ■ • ■ , tn-i not all zero such that

n-l

E t¡L' = ».

But projection is a linear operation and

E W = *" = (0, ■••• ,0)EEn.i.i-i

This means that the columns (L1)', ■ ■ ■ , (Ln~1)' of the approximate differ-

ential of g at y are linearly dependent, hence Jg(y) =0.

6. The Gauss-Green Theorem.

6.1 Definition. For/ = l, 2, • • • , re and SQEn we define the set ß,(5)

as follows:

fE®,(S) if and only if / is a numerically valued function, S is a subset of

the domain of/,

- oo < \ f(x)vj(S, X)d9x < oo(i4)( _ » < J Djf(x)dx < co,

and there is a set FC£»-i such that | V\ — 0 and

/(yi. • • • . y/-i. b, y,-, • • • , y„_i) - f(yi, • • • , y¡-i, a, y¡, ■ ■ ■ , yn-i)

= I Djfiyi, ■ • , yj-i, t, yit ■ ■ ■ , yn-i)dtJ a

whenever yEEn-i—V, a<b, and (yi, • • • , y,_i, /, y¡, ■ ■ ■ , yn-î)ES for

a = t^b.

6.2 Notation.

X" = Ei E [(y ot)EX] for X C £», y G £»_i.

Observe that in casej=w the last condition in 6.1 can be stated as follows:

/(y o b) - f(y o a) = j Dnf(y o t)dt

whenever y G£n-i— V, a<b, and [a, b]CS".

6.3 Remark. In connection with Definition 6.1 we remind the reader that

the condition

(1) - » < \ f(x)vj(S, x)d$x < »

does not imply that the exterior normal of 5 exists anywhere. In fact 3.6 tells

us that Vj(S, x) =0 whenever 5 has no exterior normal at x.

If T is the boundary(15) of S, then the relation

j f(x)d$3J m

— oo < I f(x)d$x < oo' T

is sufficient for condition (1). This follows from Theorem 4.5 and the fact that

| v(S, x)\ = l for xE En, v(S, x) = 6 for x E En - T.

6.4 Theorem. If(I) 4 is a bounded open subset of £„, B is the boundary(16) of A, and

*(£) < co ;

(II) R is the set of all points at which B is $ restricted,

C+ = A+(£„ - 4)A-(4), C- = A+(4)A-(£n - 4),

C = C+ + C-, F = En E [£*' is finite],X

and

(u) vj(S, x) is the jth coordinate of v(S, x).

(16) See footnote 10.

|(C£-£)'|^0;

(III)/£Q„U+£);then

j Dnf(x)dx = J*/(*K(4, x)d$X.

Proof. Select F££„_i in accordance with (III) and 6.1. Let

H = BE[vn(A,x) 5¿0],

W = RBE[(\i) + {- i\) dir* (73, x) = 0],X

Z = £73 £ [({¿} + {- i}) dir* (73, x) ^ 0}.

Use 5.1 to select Borel sets T\, T2, T3, ■ ■ ■ CEn-x, Lipschitzian functions

gii g2> gs, ■ ■ ■ on £„_i to En, and disjoint sets Si, S2, S3, • • • such that

Sk=git(Tk), [gk(y)]' = y for ¿ = 1,2, 3, • • • , y££„_i, and

CO

$(W - S) = 0 with S = X) S*.*-i

Next we define the set G:

xCG if and only if *££,*'£ £»_i - V, C*' £ S*'

and ft —1 2 3, • • • implies

y(4, x'ot) = Tg*(a;') whenever / £ (SÍC+)1',

v(A, x'of) = — rgk(x') whenever t £ (StC-)1',

v(^, x'ot) = 6 whenever t £ (S* - C)1'.

Let Ä* be the characteristic function of HGSk, and let

pk(x) = hk(x)f(x)vn(A, x) for £ = 1, 2, 3, • • • , x £ £„,

00

<*(y) = 12pk[gk(y)]Jgk(y) for y £ £„_i,

/%) = f #n/(y o 0<» for y £ £n_i,

The remainder of the proof is divided into eleven parts.

Part 1. If XCW with |X'| =0, then $(X)=0.Proof. X = (X-S)+2Zï=iXSkC(W-S)+2Zt"=igk*(X'), $(W-S)=0, and

gk is Lipschitzian. Use Lemma 4.1 of SA II.

Parí 2. I £„_i-£'I =0.Proof. Letting P be the projecting function such that

Pix) = *' for x G En,

we deduce from Theorem 4.4 of SA I and Remark 2.3 of SA II that

j N(P,B, y)dy g $(£) < ».

Hence N(P, B, y) < °o for almost all y in £„_i, which implies that yEF'

for almost all y in £„_i.

Part 3. \iC-WS)'\ =0.Vtoof.(C-WS)'C(C-W)'+(W-S)'C(C'-F') + (C-W)'F' + (W-S)'

= (C'-F') + (CF-W)' + (W-S)'C(En-i-F') + (CF-R)'+Z' + (W-Sy.Apply Part 2, (II), Theorem 5.5, and the relation $(W— S) =0.

Parti. *(IF-G)=0.

Proof. For each positive integer k let

Uk = £ b(y) G C+ and p(4, g*(y)) * T*»(y)]

+ f k*(y) G C- and v(4, g*(y)) * - rgk(y)]

+ f k*(y) G (F - O and *(4, **(y)) ̂ *].

In view of the definition of C+ and C~, and of the relation

(F - C) C A+(4)A-(4) + A+(£„ - 4)A~(£„ - 4),

we can apply Theorems 4.3 and 4.4 to each of the three sets whose union is Uk,

and infer

| I7*| — 0 for ft = 1,2, 3, • • • .

But it follows from the definition of G that

(W - G)' C (£n-i - £') + V + (C - S)' + ¿ Uk.*-i

Hence Parts 2 and 3 imply

\iW-G)'\=0.

Use Part 1 to complete the proof.

Part 5. \C'-G'\ =0.Proof. Apply Parts 3 and 4 to the relation (C'-G')C(C-G)'C(C-W)'

+ (W-G)'.Part 6. \A'-G'\ =0.Proof. (4'-G')C (£„-!-£') +(4'£'-C')-r-(C'-G').From parts 2 and 5 we infer that the first and third of the last three sets

each has measure zero. We shall complete the proof by showing that the sec-

ond set is vacuous.

For this purpose we suppose yE(A'F' — C'). Then 4" is the union of a


finite number of nonvacuous open intervals, and (y o sup A") £C+. Hence

y£(C'-C') = 0.Part 7. a(y) =ß(y) for almost all y in G'.

Proof. Note thatCO

£n-i — LI (domain Igk) = 0,k-X

and let y be a point for which

00

y £ G' J! (domain Jgk).k-X

Hence y££', 73" is finite, and there is a finite, disjointed family Q of non-

vacuous open intervals such that

Ay = c(Q).

LettingX=E{inf/}, P=£{sup7},

we see from the definition of C+ and C~ that

p - x = (c+)", x - p = (c-y.

Remembering that y££„_i— V, we compute:

ß(y)= E f(yot)- E f(yot) = E f(yot)- E f(yot).«Gp-x tE\-p iG(cV «£(0"

Since y CG', we know that

(c+)" = S"(c+)" = (sc+)" = E (s*c+)",t-i

OO

(c-)" - E (StC-)».*-i

Hence

ß(y) = ¿ E /(y o /) - E E f(yo t).k-1 (G (SjfcCV k-X «G (StC )»

We next use the definition of G, and 3.15, to obtain the propositions:

tC (SkC+)" implies vn(A, yot)Jgk(y) = [rgk(y)]jgk(y) = 1,

/ £ (SkC-)" implies vn(A, yot)Jgk(y) = - [rg*(y)]jgk(y) = - 1,

/ £ (S* - C)" implies vn(A, y o t)Igk(y) = 0.

Consequently00

0(30 = E E Ay o 0"n(4, y o t)Jgk(y).k-x iG(St)"


Next we notice that

VniA, y o t) = 0 for / G £i - H»,

and that y EG' implies G" = Ei. From these relations, and the identity

H«G"iSk)»=-(HGSky, we infer

ß(y) - £ E f(y ° t>ÁA, y o t)jék(y).*-l t(=(HGSk)"

Recalling the definition of hk, and the relation

(yot) = g*(y) for tE (Sk)",

we perform the summation with respect to t, and obtain

CO

ß(y) = E A*b(y)]/b(y)>»[4, «*(y)]7«*(y)Jfe-1

CO

= zZpk[gk(y)]Jgk(y) = a(y).*-l

Part 8. a(y) =ß(y) for almost all y in £„_i.

Proof. LetY = En-i E [a(y) * ß(y)],

and check that

Y C [F - (A' + G')] + YG' + (A' - G').

From Parts 7 and 6 we know that

| YG'\ =\A' -G'\ = 0.

Clearly y G (£»-i-4') implies 4" = 0 and ß(y) =0.

From the relation yE(En-i — G') we can infer Gy = 0, hk[gk(y)]=0 for

ft = 1, 2, 3, • • • , and a(y) =0.

Hencea(y)=0=)8(y)foryG[£„-i-(4'-|-C7')],and [F-(4'+G')] =0.

Thus | Y\ =0.£<zr¿9. *(iî-IF) = 0.Proof. From 3.16 we know that Df (B, x) = l for <ï> almost all x in B. Hence

4.2 enables us to conclude for <3? almost all x in H that

xER= W +Z and dir* (£, *) C £ [z-v(A, x) = 0].

But, for such a point x, the relation ï£Z implies

0 ^ v„(A, x) = i>v(A, x) = 0.

Thus x E W for $ almost all ac in H.

PartíO. $(77-GS)=0.

Proof. (II-GS)C(H-W) + (W-GS) = (H-W) + (W-G) + (W-S).Use Parts 9 and 4, and the definition of S.

Part 11. fADnf(x)dx=ff(x)vn(A, x)d$x.Proof. From Part 2 and the Fubini Theorem we infer |73| =0. Hence

v(A +B, x) = v(A, x) for all x, and (III) implies

— oo < I f(x)v„(A, x)d$x < co.

Now use Part 10, Theorem 4.5 of SA II, and Theorem 2.1 of the present

paper to compute:

I f(x)vn(A, x)d$x = I f(x)vn(A, x)d$x

= I f(x)vn(A, x)d$xJ Has

00 f%

= E I hk(x)f(x)vn(A, x)d$>x

OS /*

= E I Pk(x)d$xk-xJ

00 /»

= E I Pk(x)N(gk, £n_i, *)d**n—i«*

= E I #*k*(y)]/«*(y)áy.*=i ^Similarly

°° > f I /(*K(A x)\d$x= ¿ i I />*b(y)] I Jgk(y)dy.

Hence we may change the order of summation and integration, and use

Part 8, as follows:

J f(x)v„(A, x)d$x = J 2Z pk[gk(y)]Jgk(y)dy = J a(y)dy = J ß(y)dy.

But (III) implies

- oo < J Dnf(x)dx < 00 ,J A

and we infer from the Fubini Theorem that

j Dnf(x)dx = j ß(y)dy.

The proof is complete.

6.5 Corollary. If

(I) A is a bounded open subset of £„, B is the boundary of A, and <$(£) < oo ;

(II) Q = £„ £ [»(A, x) ^d], F = £„ £ [73-' is finite],X " X ■

C = A+(£„ - A)A-(A) + A+(4)A-(£„ - A),

and

\(CF-Q)'\ =0;

(III)/£i2„(4+£);then

j Dnf(x)dx = j f(x)vn(A, x)d*x.

Proof. Let £ be the set of all points at which B is <3? restricted, and check

that

(CF - £)' £ (CF - Q)' +(Q- £)'.

From (I), 3.16 and 4.2 we know that $((? — £) =0, and use this relation and

(II) to infer

| (CF - £)' | = 0.

Reference to 6.4 completes the proof.

6.6 Remark. With the obvious changes in the hypotheses (II) and (III)

of Theorem 6.4 (or 6.5), we obtain the formula

J Djf(x)dx - f f(x)Vj(A, x)d$x,

where/ is an integer between 1 and n.

6.7 Remark. In the conditions (I) and (II) of Theorem 6.4, the sets B, R,

C, F are all defined in terms of the set A. Hence (I) and (II) are essentially

properties of A.

It should therefore be understood what we mean by saying that a given

set A has (or does not have) the properties (I) and (II) of 6.4.

6.8 Theorem. If each of the disjoint sets Ai and A2 has the properties (I)

and (II) of 6.4, and A is such an open set that

(Ai + Ai) £ 4 £ closure (4i + 42),

then A also has the properties (I) and (II) of 6.4, and

f f(x)Vn(Ai, x)d$x + f f(x)vn(Ai, x)d3>x = f f(x)vn(A, x)d$x

whenever f is such a numerically valued function that these integrals are finite.

Proof. Recall 6.7 and 6.4, and attach the generically obvious meaning to

B, Bi, £2; R, Ri, £2; C, &, &; F, £i, £2. Since 4i and 42 have the properties

(I) and (II) of 6.4, we have

(1) *(£i) < », HBi) < 00 ,

(2) | (CiFi - Ri)' |=0, | (C2£2 - £2)' I = 0.

A simple check reveals that

(3) B C £1 + B2, CFiF2 CCi + C2,

(4) $(£) = $(£1) + $(£2) < co.

From (1) and Theorem 4.4 of SA I we infer

(5) I (En - Fi)' | = 0, I (En - F2)' I = 0,

whereas repeated application of (1), (4)and S.lóyieldsdir^^,*) = dir^ (BBi, x)

= diri,(£i, x) for f> almost all x in BBi, dir^(B, x) =dir^(BB2, x) =dir4>(£2, x)

for i> almost all x in BB2,

(6) *(££i£i - R) = 0, $(BBiRi - R) = 0.

Next we use (3) to check

2

(ÇF - R) C B(C - R) C (£CFXF2 - R) + E (C - £y)j-i

C E (5Cy -R) + (C- F,) = £ (5£;Cy - R) + (C - F,)j=i j=l

C Ë (BBiRi -R) + (Ci - R¡) + (C- Ff)3=1

2

C E {BB,R¡ - R) + (CjFj - R) + (En - Ff),

| (CF - R)' | ¿$(££J£J. -R)+\ (CFj - R,)' \ + | (£„ - £,)' |,

and we conclude from (6), (2), (5) that

1 (CF - R)' 1 = 0.

From this relation and (4) it follows that 4 has the properties (I) and (II)

of 6.4.

We see from (5) that |£i+732| =0, hence \A — (^4i+^42) j =0, and we have

f Dnf(x)dx + f Dnf(x)dx = f Dnf(x)dxJ Ax J At Ja

whenever fCün(Ax+Bx)Qn(A2+B2)Qn(A+B). For each function / of this

class we therefore know from Theorem 6.4 that

(7) j f(x)vn(Ax, *)<*** + j /(*K(42, x)d<t>x = j f(x)Vn(A, x)d<t>x.

Hence (7) holds in particular whenever/ has continuous partial derivatives

on £„, and we can apply standard methods of approximation to deduce that

(7) is valid for every numerically valued function / for which the integrals

occurring in (7) are finite.

6.9 Remark. Suppose condition (I) of 6.4 holds. Let C = A+(£„— .4)A_C4)

+A+(A)A~(E„— .4),andsupposethereareLipschitzianfunctionsgi,g2,g3, ■ • •

on £„_i to £„ such that

* C - E (range gk) = 0.

Then condition (II) of 6.4 holds, and the Gauss-Green formula is true for

every function/which satisfies condition (III) of 6.4.

Hence Theorem 6.4 includes the classical Gauss-Green Theorem.

7. The Gauss-Green Theorem in the plane. Throughout this section we

assume w = 2. Hence $> is now Carathéodory linear measure.

7.1 Definition. We say x is accessible from S if and only if #££2, S££2,

and x is a limit point of a connected subset of S.

7.2 Theorem. If A is an open subset of £2, B is the boundary of A, and x is

accessible from both A and [E2—(A+B)]t then

dJ(B, x) ^ 1.

Proof. Select connected sets a and ß such that

« £4, ß C £2 - (4 + 73),

and x is a limit point of both ctandß. Since x£73, we see that x£a, diam a>0;

x£0, diam/3>0.Let p be a number for which

0 < p, 2p < diam a, 2p < diam ß.

For r>0 we define

Cr = E2E[\z- x\ = r].

Obviously 0<r<p implies aCr^O and ßCr^O; consequently BCr has at

least two elements, because every circular arc, which joins a point of 4 to a

point of [£2— (4 +£)], crosses B on the way. Letting/ be the function on £2

to £i such thatf(z) — I g — * | for z E £2,

we infer thatN(f, B,r) = 2 for 0 < r < p.

From the well known relation(16)

$(5) = | f*(S) | for 5 C £2,

it follows, as in the proof of 4.3 of SA I, that

$(££*) ^ f N(f, BKPX, r)dr.

Hence

HBK'x) = f N(f, BK"X, r)dr = V N(f, B, r)dr = 2 (' dr = 2p = y(KÍ).J Jo Jo

Since p was an arbitrary sufficiently small positive number, the proof is

complete.

7.3 Theorem. If A is a bounded open subset of E2, B is the boundary of A,

<£(£) < 00, j is either 1 or 2, andfEti,(A +B), then

JD3f(x)dx= I f(x)vj(A, x)d<t>x.

Proof. Assuming j = 2, we use the notation of the statement of Theorem

6.4. Evidently our present hypotheses include the conditions (I) and (III).

We shall complete the proof by showing that (I) implies (II) in the present

case re = 2.

Each point of CF is accessible (by straight line segments) from both A

and [£2- (4 +£)]. Hence 7.2 implies

dJ,(B, x) = 1 for xE CF.

On the other hand we infer from 3.16 that

Di(B, x) = 1 for $ almost all x in B.

Consequently

DÍs>(B, x) = D# (£,*) — 1 for $ almost all x in CF.

Applying 3.16 once more we obtain

(16) See the proof of Theorem 4.2 of RM.

D%(CF, x) = D$(CF, x) = 1 for $ almost all x in CF.

We can now use Theorem 11.1 of RM to infer that CF is <£ restricted at i>

almost all of its points. Hence 3.16 implies

$(C£ - R) = 0.

This completes the proof of (II), and of the theorem.

7.4 Remark. We don't know the answer to the following question:

7s the analogue, for n>2, of Theorem 7.3 true or false?

The existence of Lebesgue spines shows that the strict analogue of 7.2 for

3-space is false, and we are not familiar with a proof of the 3-dimensional

analogue of Theorem 9.5 of RM.

7.5 Remark. Similar to 7.2 is the following true statement:

If ACE2, B is the boundary of A, and both A and (E2—A) have positive

2-dimensional lower Lebesgue density at x, then D$(B, x)>0.

We neither prove nor use this fact in this paper.

7.6 Remark. Using the terminology introduced in 6.7, we can describe the

idea of the proof of 7.3 as follows:

If A is a bounded open subset of £2, B is the boundary of A, and $(73) < »,

then. A has the properties (I) and (II) of 6.4.

7.7 Remark. Let p be such a continuous function on £i to £2 that — oo <s

<t< oo implies p(s) =p(t) if and only if (t — s) is an integer.

Let 73 = range p. Then £ is a simple closed Jordan curve, parametrized by p.

Let A be the set of those points of the plane which are "inside" B. Thus x£.4

if and only if xCE2, and there is such a positive number r that BCK'X and

the relation |y| ^r implies there is no continuum C for which \x] + {y} £C

CE2-B.From Jordan's Theorem we know that B is the boundary of A. The fol-

lowing facts seem of interest in case the function p is differentiate on a sub-

set of its domain :

(I) If -oo<5<oo and \p'(s)\>0, then either v[A, p(s)]-—Tp(s) or

v[A,p(s)]=rp(s).

(II) If — oo <s<t< <x>, \p'(s)\ >0, |^'(0| >0, then there is such a number

X that X2 = 1 and

v[A, p(s)] = \rp(s) and v[A, p(t)] = \rp(t).

In this connection we note that | p'(s) \ >0 implies

(-p{(s),p{(s)) ip'(s)TP(S) =

P'(s) I I p'(s)

Hence we see from (I) that a simple relation exists between the exterior

normal of A and the derivative of p, whenever the latter has a positive modu-

72 HERBERT FEDERER Lîuly

lus. We are then told by (II) that the ambiguity in sign, which is inherent

in (I), is not very serious, because the same choice of sign is correct for all points

under consideration.

The proof of (I), using Jordan's Theorem, is fairly simple. We are, how-

ever, not familiar with a short proof of (II), except for the case in which p'

is continuous and |^'(i)| >0 for — oo <s< ».

In this paper the statements (I) and (II) are neither proved nor used. One

of their immediate implications is, however, stated for completeness:

If p satisfies a Lipschitz condition, then

j f(x)Vj(A, x)d$x = + j f[p(t)]rsp(t) I p'(t) I dt

forj = 1,2, and every <£ measurable numerically valued function f.

8. Cauchy's Theorem. As in §7 we let « = 2. $ is again Carathéodory lin-

ear measure. We do not distinguish between £2 and the finite complex plane.

Hence v(A, x) is a complex number whenever 4 C£2, reG£2, and the point i,

which was fixed in 3.5, has now its classical significance. To avoid ambiguity,

we state:

8.1 Definition. If / is a function whose domain and range are sets of

complex numbers (subsets of £2), then/' is the function such that

/(«) - /(*)f'(x) = lim- for every x.

i-.x z — x

We say / is conformai at x if and only if f'(x) EE2.

8.2 Remark. If f is conformai at each point of the open set AQE2, and

|/(re)| ÛM for xEA, then xEA implies \f'(x)\ =8M/ira, where

a = inf I 3 — re .

As is well known, this follows immediately from Cauchy's integral formu-

lae for a square with center x and side a.

8.3 Theorem. If 4 is a bounded open set of complex numbers, B is the

boundary of A, "£(£) < ^> ,f is conformai at each point of A, and f is continuous

at each point of B with respect to (A+B), then

I f f(x)v(A, x)d$x = 0.

Proof. Let e>0. Determine rj>0 so that

5r[*(S) + l]i, = c,

and then select a number ô>0 such that |/(z)—/(re) | ^rç whenever xE(A+B),

zE(A+B), \x-z\ =a.

From the definition of f> we obtain open connected sets G\, G2, G3, ■ ■ ■ ,

each of diameter less than 5, for which

00 00

73 £ EG; and E t(G¡) < *(B) + 1.i-i #-1

Since B is compact, we use the Heine-Borel theorem to obtain an integer p

such that

BC2ZG,.3-1

We may certainly assume BG¡9¿0 for j = l, 2, • • • , p.

Select points x'CBG¡ and define

Cj= E2E[\z- x'\ < diam G¡], Tj=E2E[\z-x'\= diam G¡]

fori-1, 2, • • • , p. Clearly GjCC¡ and

73 £ E C¡.j-i

Now let

Ao = A- ¿(Cy+rv),j-i

k-i

Ax = 4Ci and Ak = ¿C* - E (ci + Ti) for ft - 2, 3, • • • , ¿,j-i

and denote

73,- = boundary A ¡, H¡ = £2 £ [v(A y, a;) ?* 0]

for j-0, 1, 2, • • • ,p.The theorem follows from the last of the six parts into which we divide

the remainder of the proof.

Part 1. BkCB+2Z*-iTi and $(B*)< °° for k = 0, 1, 2, • • • , p.Proof. If X+ F££2, then

Bdr (X + F) £ Bdr X + Bdr F, Bdr (£2 - X) = Bdr X.

Setting Co=A, we hence see that k=0, 1, 2, • • • , p implies

Bt = Bdr Ak £ Bdr4 + BdrC* + £ Bdr (C* + r») £ 73 + £ Ty,i-i í-ip

$(73*) ̂ HB) + 2~2* diam T,./-i

Part 2. If 0^j<k<l^p are integers, then HjHkH^O.

Proof. Otherwise we could pick a point xEHjHkHi^O and use the fact

that the set 4,-, Ak, A¡ are disjoint, together with the definition of the exterior

normal, to infer

\Krx\

1 = lim.~o+1 k: i

\kUj\ \KrxAk\ \KTxAi\ 3> lim inf —.-¡— + lim inf —.-¡—|- Ihn inf —¡-¡- = — •- ^o+ \k:\ ,-o+ \k:\ -o+ \k:\ 2

Part 3. vzZU*(Hi)=e.Proof. Let hk be the characteristic function of Hk. Then Part 2 implies

ÍZ hj(x) g 2 for x E £2.y-o

Use this relation and Part 1 to check:

E *(#/) =ÍZf h,ix)d*x = 2$( ¿ h) = 2*( E 5,)i-o y-o *> \ y=o / \ ;-o /

g 2$( 5 + Ery) = 2*CB) + 2irX)diamry\ j-i / 3-=i

= 2*(£) + 4x E diam Gy = 29(B) + 4x X) 7(Gy);'-l ¿«1

< 29(B) + 4ir[*(£) + 1] = 5ir[$(B) + l].

Multiply by 17 to complete the proof of Part 3.

Part 4. \ff(x)v(A,-, x)d$x\ ^r]$(Hj) forj = l, 2, ■ ■ ■ , p.Proof. From the relation HjEBjEC, + T¡ it follows that xEHj implies

j*—re'l =2_1diam C,- = diam G¡<5; hence

I f(x) - f(x') I = v for x E Hj.

Since <f>(£,) < 00 by Part 1, we know from Theorem 7.3 that

I v(A¡x)d$x = 0,

I f(x)v(A¡, x)d$x = I [/(*) - /(jci')]"(4;, re)á$rc +/(reO I v(A¡, x)d9¡

and conclude

/ I*x) - f(x>") I d$x = r,9(Hf).

1945] • THE GAUSS-GREEN THEOREM 75

Part 5. ff(x)v(A0, x)d$x = 0.Proof. Let

M= sup \f(x)\,

b = inf | x — z |.«G^o.'G-B

Since/ is continuous on the compact set A +B, we have M< oo. On the other

hand

73(closure 4o) £ (£ Cj)(E2 - £ Q) = 0,Í-1 3-1

the compact sets B and (closure A0) are a positive distance apart, and &>0.

Hence 8.2 implies

|/'(*)| = &M/wb for xCAo.

Let /i and /2 be such functions that

f(x) = (fi(x),f2(x)) for every x,

and recall the Cauchy-Riemann equation

(Dxfx(x), Dif2(x)) = /'(*) = (Dif2(x), - D2fi(x))

for xCA.

Consequently

| Djfk(x) | = | /'(*) | ^ 8M/*b for x £ 40; / - 1, 2; ft - 1, 2.

It follows that

{/i} + {/2} £ fli(4o + 73o)02(4o + 73o).

Using Part 1, the last relation, and Theorem 7.3, we compute:

I f(x)v(A0, x)d$x = I [fi(x)vi(A0, x) — f2(x)v2(A0, x)]d$x

+ *J [fi(x)v2(Ao, x) + fi(x)vx(A0, x)]dx

= f [Dxfx(x) - Dif2(x)]dxJA0

+ i f [Difx(x) + Dxfi(x)]dx = 0.JA„

Part 6. \ff(x)v(A, x)d$x\ =e.Proof. The sets Ao, Ax, -42<, ■ • • , Ap are disjoint open sets whose bound-

aries have finite <$ measure and for which

76 HERBERT FEDERER

E 4 y C 4 C closure 53 4 y.y—o y—o

Hence we use 7.6, 6.8, and Parts 5, 4, 3 to conclude

( f(x)v(A, x)d$x = I E f/(re)v(4y, x)d$xJ i—oJ

á ¿I ( f(x)v(Ai,x)d*iy-o IJ

= E n*(Bt) á *.y-i

8.4 Remark. The hypotheses of 8.3 imply

1 C f(x)v(A, x)fit)-I •"■' d$x for z G 4.

2x J x — z

This Cauchy formula is proved by the standard method, except that cross-

cuts are unnecessary (here, and in similar situations), because Theorem 8.3

applies directly to the open set obtained from A by removing a closed circular

disc with center z.

8.5 Remark. If p is a Lipschitzian function of the type described in 7.7,

and if the hypotheses of 8.3 are satisfied, then

if f(*>(A, x)d*x = + j f[p(t)]irp(t) I p'it) I dt

= ±!of[p{t)]\mlP'it)ldt

= ± f f[p{t)]p'(t)dtJ o

= + §¿[p{t)]dtp(t),

the sign depending on the sense in which p parametrizes B.

Hence Theorem 8.3 includes the well known strong form of Cauchy's Theo-

rem for a simple closed curve.

University of California,

Berkeley, Calif.


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