lebesgue area and hausdorff measure
TRANSCRIPT
205
LEBESGUE AREA AND HAUSDORFF MEASURE
by Earl J. Mickle (Columbus, U.S.A.)
1. - I N T R O D U C T I O N .
I.I . Let
(1) T: x, = x,(u, v) x2 = x2(u, v), x3 = x3(u, v), (u, v )~Q,
be a continuous mapping from the unit square Q: 0 ~ u - ~ l , 0 ~ v ~ l , in
the uv-plane into Euclidean xlx2x3-space $3 and let A(T) denote the Lebesgue
area of the surface of the type of the 2-cell represented by T. The reader is
referred to Rad6 [14] for background material and definitions (square brackets
refer to the bibliography at the end of this paper). One of the basic problems
in the study of surface area has been to introduce a suitable multiplicity func-
tion and a suitable measure so that A(T) is given by integrating the given
multiplicity function with respect to the given measure. The reader is referred
to Federer [8] for an exellent discussion of this problem. Let H~(E) designate
the Hausdorff 2-dimensional measure of a set E C S~ (see w 2.2 for the precise
definition), let x designate a point in $3 and for a set E C Q let N(x, T, E) designate the number (possibly --[- co) of points in E A T -l(x). Then a suitable
choice of a multiplicity function and measure might be given by
=fN(x, T, Q)dH ~. AH(T)
However A ( T ) ~ A~(T) and simple examples show that in general the equality
sign does not hold. Special cases where the equality sign holds have been
given by Cesari [3], Besicovitch [l], Federer [7], Mickle and Rad6 Ill] and Rad6
and Reichelderfer [16].
2 0 6 E A R L J. M I C K L E
For xES 3, T - ' ( x ) ~ ~1, a component of T-'(x) is called a maximal model continuum (m.m.c.) for 7". For E C Q let N*(x, T, E) designate the number of
m.m.c. "s under T that intersect E ~ T -1(x). Then a suitable multiplicity func-
tion and a suitable measure might be given by
a~(r) = fN*(x , T, Q)dH ~.
However, as in the case of AH(T), A(T)~< A'~(T) and simple examples show
that in general the equality sign does not hold.
1.2. At a point (u, v) E Q where the partial derivatives of the functions
defining E in (1) that are involved exist we define the Jacobians
J, (u, v) = x~o x ~ - x ~ x ~ , ]~ (u, v) = x~o x , , - x,~ x ~ , . /3(u, v) = x,~ x~, - x ~ x , , ,
I
j (u, v) = [Z + ]~ + J~]~.
For a plane set E let L~(E) designate the Lebesgue exterior planar measure
of E. If B0 is a Borel set in Q such that J(u, v) exists everywhere on B0,
Federer [5] has shown that
(2) /J(u, v)dL, = f N(x, T, Bo)dH'. Bo
If J(u, v) exists everywhere on Q except on a set of L,-measure zero, is sum-
mable on Q and
(3) A(T) = fJ(u, v)dL~, Q
then a Borel set B oC Q can be found so that J(u, v) exists everywhere on Bo,
L 2 ( Q - B o ) = 0 and for which by (2) and (3),
A(T) = fN(x , T, (4 ) Bo) d H ~ .
Cesari [4] has shown that every surface of the type of the 2-cell of finite Le-
besgue area has a representative T for which J(u, v)exists everywhere on Q
except on a set of L2-measure zero, (3) holds and hence for which there is a
Borel set BoC Q such that J(u, v) exists everywhere on Bo, L 2 ( Q - Bo)= 0 and (4) holds.
1.3. Federer [6], Heisel and Rad6 [9], Mickle and Rad6 [10] have shown
that a suitable measure and (by t-deformations of a mapping) a suitable multi-
plicity function can be obtained so that A(T) is given by integrating the given
LEBESGUE AREA AND HAUSDORFF MEASURE 207
multiplicity function with respect to the given measure. However, in each of
these cases, the measures are not over sets in $3. In this direction Feder [8]
introduces a multiplicity function L~(x, T, Q) by an averaging process and
shows that A(T) is given by integrating L~(x, T, Q) with respect to the
measure H 2. It is the purpose of this paper to contribute to the lines of inve-
stigation given above by establishing the following result. Among the m.m.c.'s
under T certain ones will be defined to be essential maximal model continua
(e.m.m.c.) under T. We show that if K(x, T, Q) designates the number
(possibly ~ ) of e.m.m.c.'s under T in T-~(x) then
A(T) T, Q)aH ~.
2. HAUSDORFF AND SOME ALLIED MEASURES.
2.1. For a set E CS3 we denote by d(E) the diameter of the set E and
for xES3, r > 0 we denote by s(x, r) the open sphere with center at x and
radius r.
2.2. For a set E C Ss. Let ~ be a generic notation for a finite or denume-
table number of sets e~ e2, .. which satisfy the conditions (i) E C 1.3 ej and (if) for
each j, d(ej) < E, ej ~ ~ and let x~ be a generic notation for a finite or denu-
merable number of open spheres sl, s2 . . . . which satisfy the conditions (i)
E C U sj and (if) for each j, d(sj)~ ~. We define
H~(E) = gr.l.b.~7~/4d(ej) 2 for all ~ ,
S~(E):gr.l.b.Z7~/4d(s) 2 for all -c~ j J �9
Since the set function H~(E) increases when e decreases, it has a limit (finite
or infinite) for r ~ 0. The Hausdorff 2-dimensional measure of E is defined to be
H2(E) = lim H~(E). ~.~o
H2(E) is a measure over a completely additive classe of sets I' in $3 which
contains all open sets, all closed sets, all Borel sets and all analytic sets.
2.3. LEMMA. For any set E C $3, ~ > O, S~,(E)~< 4H2(E).
Proof. For E C S 3 , e > 0 , let e > ~ > 0 be given. Let el, e 2 . . . . . be a ~
and let ~ be such that 0 < ~ < 8, (1 + ~ ) d ( e j ) < ~ , j = 1, 2 . . . . . Let x" be
a point in ej and let sj be the open sphere with center at x i and radius
208 E A R L J. M I C K L E
r j = max [(1 + 5i)d(ej), a/2J+~]. Then E C U e j C j U s j and d(si)<.2s. Thus
(1) S~(E) ~< Ejrc/4d(sj) 2 ~< 4(1 + ~)2~r~/4d(ej) 2 + 82~/4 for all ~,.
The relation (1) implies that S~ (E)~4 (1 + ~)2 H2 (E)_[_~ ~ / 4 ~ 4 (1 + ~)2H2 (E) + ~2 n/4.
Since ~ > 0 is arbitrary, it follows that S]~(E)-~< 4H2(E).
2.4. For a set EEl" and xES3 we define
h (x, E) = lim sup H 2 [E C) s (x, r)]/n P. r - ]~o
2.5. LEMMA. I f E ~ l", Eo = [x ] x E E, h (x, E) = 0], then H 2 (Eo) = 0.
Proof. Case 1. H2(E) < co. For each pair of positive integers n, m we set
E , , , ~ = [ x i x ~ E , H 2 l E n s ( x , r ) ] ~ n d / n for 0 < r < l/m]. ~-
Then Eo = ~ ~A E .... C ~ E3,,,~ and it is sufficient to show that H*(E~2,,~)= 0 n ~ l m = l m = l
for each integer m. For a given m set 2E ----- l/m. Let s;, s~, . . . be open spheres
such that E~,m C U s;, for each j, cl(s~ < 2~ and there is a xJEE3~,,~ A s.;. For
rj = d(s~,
(2) H~(E32,~) ~< EH2[E32,., N s(x j, rj)] ~ 1/32Y, nry = 1/8~(~/4)d(s~ ~. J j 1
From (2) and the lemma in w 2.3, 2 !
(3) H~(E32,~,) ~< 1/8S2,(Es2,m) ~< ~-H"(Es2,,,).
2 Since H (s2,,,) < co, (3) implies that H2(E32,m) = O. Case 2. H~(E)= co. Set
E t = [ x l xEE, H ~[ENs(x , I ) ] = co for r > 0 ] .
Since h(x, E ) = co for x EEt, E o C E - E 1. Since E - E~ is the union of a
denumerable number of sets of finite HLmeasure , the fact that H*(Eo)= 0 follows from Case 1.
2.6. LEMMA. I f E ~ l", H ~ (E) < ~ , Eo = [x [ x ~ E, h (x, E) > 0], then H ~ (Eo) ----- 0.
Proof. For each positive integer n let
(4) E, = (x ix ~ E, h (x, E) > 1/n].
Then E o = ~ E, and hence it is sufficient to show that H ~(EJ = 0 for each n.
Let ~ > 0 be given. Then E~l", H~(E) < co implies that there is a closed set
F C E such that H~ F) < ~. Then for the sets E, defined in (4)
E, = [x lx$E, h(x, E - - F) > l/n].
L E B E S C . U E A R E A A N D H A U S D O R F F M E A S U R E 209
For s > 0 given and x E E , there is a sequence of open spheres s(x, r'~) such
that r~, < , / 1 0 , r ~ . ~ 0 for m . ~ o o and H ~ [ ( E - - F ) O s ( x , r'.,)]>~r~/n. These
open spheres cover E. in the sense of Vitali and hence there is a disjoint se-
quence s(x i, q), s(x "z, r ~ ) . . , of them such that E . C U s(x i, 5 0 (see Morse J
[12], w 3.10). Thus
= 25X, r ..< 25nXH [(E - - n s(x' 0 ] (5 ) J i '
~< 25nH2(E - - F) < 25n~
(5) implies that H2(s ~< 25n~ and since ~ > 0 is arbitrary, H2(E,) = O.
2.7. Let �9 be a measure over the classe of H2-measurable sets ~ such that
for EEF there are sets E~, E~EF satisfying the following conditions (i) E ~ E ~ U E ~
E~ f~ E~ = ~. (ii) �9 (E) = @ (E~) = H ~ (E~). (iii) �9 (E~) = 0. Set
~o(x, E) : lira sup O[E N s(x, r)]/~f z. r - .~ - o
LEMMA. For EEF, E o : [x lxEE, ~r E ) > 0] we have that (a) EoEI',
(b) �9 (E) = �9 (Eo) = H e (Eo) and (c) �9 (E - - Eo) = 0.
Proof. It is easily verified that for each pair of positive integers n, m the set
E,,m = [x I x E E, �9 [E N s (x, r)] >1- 7c r2/n for 0 < r < 1/m] is closed with respect
to E and hence E,,,~ E F. Since E - Eo : ~ ~ E,,,~ it follows that ( E - Eo)6I' n ~ l m = l
and hence EoEF. Thus (a) holds. To prove (b) and (c)we shall have to consider
several cases.
Case 1. O(E) < co. It follows easily that for the sets E~, E~ given above
that H ~ (E') = �9 (E') for E ' C E~, E' E 11. Hence ~ (x, E) ~ ~o (x, E~) = h (x, E~).
By the lemma in w 2.5, H ~ ( E ~ - - E o ) : O and by lemma in w 2.6, H~(Eo--E~)=O.
Thus
(6) H2(Eo) = H2(Eo-- E~) + H2(Eo n E~) = H~(eo N E~) + H~(E~-- Co) = H~(E,) = O(E),
(7) O(E - - Eo) = O[(E - - Eo) A E,] : O(E~ - - Eo) --< H2(E~ - - Eo) = O.
From (6) and (7)i t follows that O ( E - E o ) = 0 and * ( E ) = * ( E o ) = H2(Eo).
Case 2. * ( E ) = c ~ , E = ~ E , , E,,EF, * ( E ~ ) < o % n = l , 2 . . . . . By case n = l
1, for the sets Eo : [ x l x E E , , ~(x, E , ) > 0], O ( E , - E o ) ~ 0 and since oo
C E - - E o U(E,--Eo), O(E--Eo):O. Thus (I)(E)=*(Eo) and since He(Eo)>~O(Eo)�88 n = l
H (Eo) = r (Eo). Case 3. O ( E ) = co. Set
E~ = [x lx r [E f-) s (x, r)] = oo for r > 0], E, = E - - E,. 1~ _ R e n d . C i r c . M a t e m . P a l e r m o - - S e r i e 11 - T o m o I V - 1 9 5 5
2 1 0 ~ a ~ J. M | C K L I g
It is easily verified that Ei is a closed set. Hence, since Et N E~=I~, q~(x, E ) =
q~(x, E=) for x EE2. Since E2 is the union of a finite or denumerable number of
sets in r of finite O-measure, for E ~ = [xlxEE~, ~(x, E~)> 0]-----E2NEo it
follows from case 2 that �9 (E~ - - Eo) = 0. Since ~ (x, E) = co for x E E f3 Et it
follows that E - - E o - - - E 2 - E ~ and O (E - - Eo) ----- 0. By the same reasoning
as used in case 2 �9 (E) = �9 (Eo) = H2(Eo) ~- co.
2.8. Let r o be a subclass of the class of H2-measurable sets P that sa-
tisfies the following conditions: (it E E r , EoEFo, E C E o implies that E E F o.
(if) E E P implies that there is an E o E r o such that Eo C E . (iii) E, EPo, n = 1, 2 . . . .
implies that (U E,)EP. For E E r we set
(8) O(E) ----- g r . l . b .H~(E- - Eo) for sets E o E r o.
LEMMA. The set function �9 (E) defined in (8) is a measure on the class of
sets r and satisfies the following conditions: (at E E r o implies that ~ ( E ) = O.
(b) E E P implies that there is a set E, EP o. E, C E such that for g , ~ E - - E,,
,I, ( ~ = ,I, (Eo) = H ~ (G).
Proof. (if) implies that O(E) is defined for E E P and O(E) is obviously
non-negative. It follows from (8) that O(E) ~ 0 for E E P o. From (iii) it follows
that there is an E , ( r o , E, C E such that for E~----E--E, , t~(E)=O(E~)=H~(E~).
Thus (at and (b) are satisfied.
Let E,, e r, n = l, 2 , . . . , E, n Ej = fJ for i ~ j, E = U E,,. For Eo C E,
Eo E Po, n~ (E - - Eo) = H e (E, - - E, N eo) -]- H 2 (e 2 - - E, N Eo) q-- �9 �9 �9 ~ (I) (E~) -+-
~(E2) q- ' ' ' . Hence
n
For eoC G , G~ro , n = l , 2 , . . . , H~CG--E'o)+~CG--Eo ') + . . . . H~[u(~ _ e o 3 ] = /1
H ~ [(E - - U E~)] ~ (1) (E). Hence II
/I
(9) and (10) imply that O(E) is a measure for sets E E r .
2.9. Let n~, ~ , 7:3 be the orthogonal projections of $3 onto the x2x3-plane ,
x~x~-plane, x~x2-plane respectiveIy. For the class of H~-measurable sets r we set
r, - - [ e l t r e r , z . ,~,(~ - - o], i = ~, z, 3, r , = r , n r , , r~ = r , l n r~,
LEBESGUE AREA A N D I~kU-qDORI~ MEASURE 211
Each of these classes of sets satisfies the condition of the lemma in w 2.8.
Hence
H l ( E ) = g r . l . b . H Z ( E - - E l ) for E, Er , , i ~ - 1 . . . . , 7,
are measures over the class of sets EEI' . EEP t implies that H i ( E ) = 0, EEP
implies that there an ~ E P , , E~ C E such that for ff~ = E - - E~, Hi(Eta) = HZ(E~,).
We also have the relations that H~(E)>~L,%(E) for i = 1, 4, 5, 7, HI(E)>~L2~(E)
for i ~ 2 , 4 , 6 , 7 a n d HI (E) >t L2 ~3 (E) for i = 3 , 5, 6 ,7 . For EEI', xES 3we set
h,(x, E) = l i m s u p n l [ E n s ( x , r)]/zcr ~, D;(E) = [ x l x E E , hi(x, E) > 0]. r-~O
By the lemma in w 2, 7, EEl ' implies that O;(E)er, H,[D;(E)]=H,(E)=H~[D;(E)] and H~ [E - - D; (E)] = 0.
3. ESSENTIAL MAXIMAL MODEL CONTINUA.
3.1. Let
T: x = xl (u, v), x , = x , (u, v), x , = x,(u, v), (u, v)e Q,
be a continuous mapping from the unit square Q: 0-.~<u-~<l, 0-~<v-~<l into $3.
For the orthogonal pr%z:~ions ~t, ~2, % of $3 onto.lthe x2x3-plane, x3xl-plane,
xtx2-plane respectively we set
TI = = t T: X~= X'(U' V)' T 2 = n , T : x3 : x~(u, v), x,=x~(u, v), x,=x,(u, v),
For i : 1, 2, 3 let E~ be the union of all
T 3 = % T: x, = xt (u, v), x2 = x2 (u, v),
(u, v ) eQ .
the essential maximal continua
(e.m.m.c.) under the plane mapping Tt (see Rado [14], pp. 281-282). We define
following sets.
e~=E~--(E~UEs'), e2=E~--(E;UE~, e~=E~--(E~UE~,
e I = ( E ; A E ~ - E ; , e ~ = ( E ; M E ~ - - E ; , e ; = ( E ; M E ~ - - E ; , e;-----E~AE;ME;.
Each of the sets el, . . . , e~ is the union of m.m.c.'s under T, e~ne~ = 0 for
i # j and e[O . . . Ue;=E~UE~UE~. 3.2. An m.m.c, y under T will be called an essential maximal model con-
tinuum (e.m.m.c.) under T if there exists an i, I -_< i.-.< 7, such that (see w 2.9)
TCe~ and T(T)ED;(T(e~), i.e., if T C e ; and ht[T(T), T(e~] > 0. For E C Q we
set K(x, T, E) to be equal to the number (possibly + oo) of e.m.m.c.'s T under
Tsuch that TAEMT - t ( x ) # r
The definition of an e.m.m.c, is used in two different senses in the pre-
ceding definition and the definition for plane mappings. For T as given in w 3.1
212 E A R L J . M I C K L E
assume that x 3(u, v ) = 0 . Then E ~ = E~=~I , e ; = l J for i = 2 , . . . . 7 and
e ~ = E~. Then (see Rad6 [14], pp. 292-293) T(e~)= TI(E~)is an open set in
the plane n1($3) and hence D~[T(e~)] (q T(e~)= T(e~)= Tt(E~). Thus the two
definitions are equivalent for plane mappings.
3.3. For T as given in w 3.1, let T----lm, m: Q->-M, 1: M->.S3, be a
monotonelight factorization of T (see Rad6 [14], pg. 53) where m is a continuous
monotone mapping from Q onto a metric space M and l is a continuous light
mapping from M into $3. Since E~, E~, E~ are Borel sets (see Rad6 [14], pg. 296)
and the union of m.m.c.'s under T, e~, . . . , e; are Borel sets in Q and
m(e~),.. . , m(e;) are Borel sets in M. For E C M let N(x, l, E) denote the
number of points in EN l-t(x). By a well known result in measure theory
N*(x, T, e~)= N(x, 1, m(e~)) is an Hrmeasurable and an H2-measurable func-
tion. Since D~(T(eD)EF,
(1) K(x, T, e;)= I N* (x, T, e~) for x E D~ (T (e~) ),
0 for xSD;(T(e~),
i s a non-negative H~-measurable function. Thus
(2) K(x, T, Q ) = K ( x , T, e;) + . . . + K(x, T, e;)
i s a non-negative H~-measurable function.
LEMMA. We have the following relationships
(3) fK(x, T, e;)dH~= fN*(x, T, e~dHl, i = 1 . . . . . 7,
(4) fK(x , T, Q)dH~ fN*(x, T, e:)dH, + . . . + fN*(x, T, e;)dH,.
Proof. All the integrals involved in (3)and (4) exist either finite or infinite.
Since (4) follows from (3) and (2) we need to prove only (3).
Case 1. Hl[O;(T(e;))] =H~[D;(T(e;))] = co. Since K(x, T, e ; )>~l and
N*(x, T, e~)>/ 1 for xED~(T(e~) both of the integrals in (3) for the given i are co.
Case 2. Hi[D;(T(e~) ] = H~[D;(T(e~)] < co. For EED;(T(e~), EEF, we have
that H i ( E ) = H~(E). Thus, since Hi[T(e~--O;(T(e~)] = 0 (see w 2.9), for
D; = D;(T (e;))
(5) f T, e~dHt = f N*(x, T, eDdHi ----- f N*(x, T, e~) d H 2 . D~ D i
(3) thus follows for the given i from (1) and (5).
LEBE.~GUE AREA A N D HAUSDORFF MEASURE 213
3.4. LEMMA. If T as given in w 3.t is such that A(T) = co then
A(T) =fK(x, T, Q ) d H ~.
Proof. A ( T ) = co implies that
f A r* (x, T,, ~ ) d Lz = co (6)
for i = 1, 2, or 3, say i = 1 (see Cesari [2] and Rad6 [13]). Thus, from (6) for
i = 1, the Lemma in w 3.3 and the remarks in w 2.9 7
fK(x, T, O)aH~= ~fN*(x, T, e;)dH,>~ i=1
>~ ~ fN*(x, T, e)aH, >~ fN*(x, T,, g;)dL~ = co. i=1 , 4, 5, 7
3.5. LEMMA. Let E be a Borel set in E~ U E~ U E; such that Lz T~ (E N E~)= 0
for j = 1, 2, 3. Then T(EGe~)EF and Ht[T(ENe;)] = O for i = 1, . . . , 7.
Proof. Since E is a Borel set, E O e ~ is a Borel set and T ( E A e l ) E I ~,
i = 1 , . . . , 7.
Case 1. i = 1, 2 or 3. Then EAe~CEAE~ and L ~ t T ( E A e ) = L2 Ti(Ene~)~<
LzT , (EN E;) = 0. Thus T ( E A e;)~P, and H , [ T ( E A e)] = 0.
Case 2. i = 4 , 5 or 6. Then there are integer il, /2 such that 1 ~ i 1 < i ~ < 3 and r , - = Fe, n Pc=. Then e~C El, n E~2 and L=% T(ENe~) = L2T~n(EA e) ~< L ~ T , . ( e n ED = o, n = 1, 2. Thus T ( e n 4 ) e r , and Ht[T(EAe;)] = O.
Case 3. i = 7 e; = E~ n E~ n E~, rr = I'~ n I~2 n Pa and L2~ T(E n e;) =
L,T,(EAe;)~<L, T , ( E A g ) = 0 , j = l , 2, 3. Thus T(ENe;)EP, and I-IT[T(EAe;)]=O.
3.6. A continuous mapping T as given in w 3.1 is called eB V (essential
bounded variation) if
fN*(x, Ti, E ) d L ~ < co for j = 1, 2, 3,
and is called eA C (essential absolute continuity) if it is eB V and ifor every
set E of L2-measure zero in Q, L 2 T i ( E N E ) = O , j = 1, 2, 3 (see Rad6 [15]).
Let E * = e ; U . - . U e ; , let E be the union of the m.m.c.'s under T in E*
that reduce to a single point and let E=, E2, Ea be the union of the e.m.m.c.'s
under 7"i, T=, T a respectively that reduce to a single point.
LEMMA. If T is eA C then T[(E* - - E) n 4]Er and H~[T(E* - - E) n e~))] = 0
for i = l, . . . , 7 . Proof. It is easily shown that E*- -E is a Borel set and hence T[(E*--E)Ae)]EP
for i = l , . . . , 7. It follows from Rad6 [15] that L 2 Ti(~--Ej)=O for j = 1 , 2, 3. Since
2 1 4 , , , t , , , , J', M I C K L E
(E*-- E) O E~j C E)-- Ej, it follows that L, TIt(E* - - E) O E~J--.<L~ Tj(E~-- Ej) = 0 , j = 1, 2, 3. By the lemma in w 3.5, H,[T((E* - - E) O e))] = 0, i = 1, . . . . 7.
3.7. LEMMA. Let T as given in w 3.1 be eA C. Then, for i = l, . . . , 7,
(7) f N*(x, T, eDan, = f N(x, T, e?dH,.
Proof. N*(x, T, e~)-.< N(x, T, e3, for x ES3. Set
E, = [xilN*(x, T, e3 < N(x, T, e3].
Then EIEI' and (7) will follow if H t ( E ) = O. Now xEE~ implies that there is
an m.m.c, under T in e~n T- ' (x) that does not reduce to a point. Hence (see w 3.6) E,C T [ ( E * - E ) n e~] and by the l emma in w 3.6, H , ( E ) = O.
3.8. LEMMA. Let T be a continuous mapping as given in w 3.1 for which the
Jacobian J (u, v) exists everywhere on Q except on a set of L~-measure zero, is summable on Q and
A(r) = fJ(a, v)aL,. q
I/ 13, is a Borel set such that J (u, v) exists everywhere on Bo and L, (Q -- Bo) = O, then 1-~ [T(Bo n e,')] = 1-1, [T(Bo n e;)] < oo, i = 1, . . . . 7.
Proof. We have (see w 1.2)
m[r(Bo)] -.< f N(x, T, Bo)dH ~ = fj(u, v)aL, < co. B o
Let Bt be a Borel set such that B, CT(BoOe~, B, EP,. Then B~----- T-'(B,)A e~OBo is a Borel set and T(B~)= B~. Then
/-/~(B,) -.< f N(x, T, B)dl - f 2 = f j(u, v)aL, B;
(8 ) 3 a
flL(u, v)laL,----~-~ N(x, T,, B;OE',)eL,. ..< n=: B; ~='
Now each of the three integrals at the end of (8) is 0 either because N(x, Tn, B;A~)-- - -0 or because L, T n ( B ) = 0 . T h u s / - / 2 ( B ) = 0 . Since H~[T(BoOe)] < oo it follows that /-F2(E~)= 0 for E~C T(BoOe~), E~EP,. Thus
/-/2[T(B0 N e~)] = H,[T(B o O e,')] < oo for i = 1, . . . , 7.
3.9. THEOREM. Let T be a continuous mapping as given in w 3.1 for which the Jacobian J (u, v) exists everywhere on Q except on a set of Lcmeasure zero, is summable on Q and
A(T) = fJ(u, v)aH,.
L.EBESGUE AREA AND HAUSDORFF MF_.ASUItE 215
Then
A (7") = f K (x, T, Q) d m. (9)
Proof. There is a Borel set Bo such that J(u, v) exists everywhere on Bo
and L , ( Q - B o ) = 0. Then (see w 1.2)
(10) A(T)= fJ(u, v)dL,---- fN(x , T, Bo)dHE Bo
For B = B o-(e~U . . . LJe;), N(x, T~, B A ~ ) = 0 for n----l, 2, 3 and hence
3
f N(x, r, B)aw = f j(u, v)aL,-.< X f lJ.(., v)ldL, B n = l B
(11) s
~E IN(x, T,, B N E'.)dL, = O. n=l
From (11) it follows that
7
(12) fro(x, r, eo)am= fm(x, r, BoN(e:t.3 ... Ue;))dH'=~fN(x, T, BoNe;)dI-I z.
For E = (Q - - Bo) (q (El LI E~ LJ E;), L, (E) = 0 and since T is e A C (see Rad6
[14], pp. 502-503) L, Tn(ECqE',)=O, n = 1, 2, 3. By the lemma in w 3.5,
H~[T(E(~ e;)] = 0 for i = 1, . . . , 7. This fact and the lemma in w 3.8 gives us
7 7 7
(13) E fN(x, r, e;)dn,= X fN(x, r, Bone;)an,= 2 fN(x, T, Bone,~an2. l ~ l t = 1 l = 1
(9) then follows from (10), (12), (13), the lemma in w 3.7 and the lemma in
w 3.3.
4. FRI~CHET EQUIVALENT MAPPINGS.
4.1. For a metric space M we denote~by d(y,, Y2) the distance between
two point y, and Y2 in M and by d(A, B) the distance between two sets A
and B in 34. For a point y in M and a real number r > 0 we denote by s(y, r)
the set ot points y ' in M for which d(y, y') < r. 4.2. LEMMA. Let T, T r be two Frdchet equivalent continuous plane mappings
(see Rad6 [14], pg. 57) from Q: 0 ~ _ ~ u ~ l , o - ~ v ~ l , 0 : 0 ~ u - ~ 4 , 0 ~ v ~ l
respectively into the xlxz-plane $2 and let
T = l m , m: Q-~M, l: M-~S2; T = l m , m: Q-~M, l: M-~S~,
216 ~.ARL ~. ~,ICKL~
be simultaneous monotone-light factorizations of them such that m and m are
Fr~chet equivalent (see Youngs [17], pp. 329-330). Then an m.m.c. T under T is
an e.m.m.c, under T i f and only i f m- lm (7) is an e.m.m.c, under T.
Proof. If 7 is an m.m.c, under T then T = m -1re(T) is an m.m.c, under T--.
Hence it is sufficient to show that if T is an e.m.m.c, under T then T is an
e.m.m.c, under ~ Assume 7 is an e.m.m.c, under T. Let (~o be an open set
such that T C O o C Q ~ where 2 ~ is the interior of Q. Let O be the union of
all the m.m.c.'s under T that lie in 60. Then O, re(O) are open sets and
T C O C O o C Q ~ For y = m ( y ) = m ( y ) , set ~ = d [ y , M - - m ( O ) ] > O . Then
0 = m -~[s(y, ~/2)] is an open set such that "( C O C QO. Since T is an e.m.m.c.
under T there exists a Jordan region/? with interior R ~ such that T C R ~
and /? is an indicator region for T(T) under T (see Rid6 [14], pp. 281-282).
Let w, w represent points in Q and Q respectively. Since m and m are Fr6chet
equivalent there exists a sequence of homeomorphisms ~v= h~(w) from Q onto
such that d[m(w), mh~(w)] < 1/n for wEQ, n = 1, 2, . . . . We assert that there is an n o such that for n > n o
a) T C h~ (/?o) C h~ (R) C O,
b) h~(/~) is an indicator region for T(T) under T,
and hence that T is an e.m.m.c, under T. Now there exists an n~ such that for n > n~ (b) holds. Since the union of the m.m.c.'s under T that lie in R ~ is
an open set which contains T and whose image under m is an open set,
d[y, m(Q - - R~ > 0. Hence (b) holds for n > no where
no = n~ + 2/~ + 1/d[y, m (Q - - R~
We first show that h~(R) c O - for n > n o . Now w~R, hn(w)~(Q - - O)
implies that d [y, m (w)] < ~/2, d[y, m hn (w)] > ~ and hence d[m (w), m hn (w)] >~
d[y, mh~(w)] - - d[y, re(w)] > ~ - - ~/2 = ~/2 > 1/n o . This is a contradiction
and hence h~ (R) C ().
Finally we show that T-Ch~(/~~ Now w E Q - R ~ h~(w)ET implies that
d[m(w), mh~(w)]=d[m(w) , y]>~d[m(Q--R~ y] ~ 1~no. This is a contradiction
and hence T-Ch~(R~
4.3. Let T, T be two Fr6chet equivalent continuous mappings from Q:
0 ~ < u ~ l , 0 ~ v ~ < l , Q: 0 - ~ < u ~ l , 0 - ~ < v ~ l respectively into Ss and let
T = l m , m: Q ~ M , I: M-~S3, 7"=Ira , m: Q-~M, l: M~.S3 ,
be simultaneous monotone-light factorizations of them such that m and m are
L E B E S G U F ~ A R E A A N D H A U S D O R F I ~ M E A S U R E 2 1 7
m m
Fr6chet equivalent. Let e~, . . . , e; and e~, . . . , e; be the sets defined in w 3.! with respect to T and 7 ~ respectively.
LEMMA. Under the above conditions e; ~ m-l m (e;), T(e;) ~ T(e;), N*(x, T , e;) =
N*(x, T, e;) for i = 1, . . . , 7.
Proof. To prove the statements of the lemma it is sufficient to show
that if T is an m.m.c under the T and TCE~ for some j, 1 -~< j~<3 , then
= m-lm(,;) C E~ (since y is an m.m.c, under 7). Let
~ l -~- Itmj, mr: M ~ M t, It: Mj-~ ~j(S3),
be a monotone-light factorization of %.I. Then
Tj ---- 7~tT = l~(mtm), mtm: Q-~Mj , It: Mt->.7~t(S3),
= %.7" ~ lt(mtm), mjm: Q ~ . M t , lj: Mt-~.7~t(S3) ,
are simultaneous monotone-light factorizations of Tj and Tj such that, since m
and m are Fr6chet equivalent, mjm and mjm are Fr6chet equivalent. Hence, if "f
is an m.m.c, under T and T C E~ there is an e.m.m.c T* under T t such that
y C y * C E ~ and hence y m- lm(y) C - - I -1 ~, = m mj mtm(y* ) ~- and by the lemma
in w 4.2, y* is an e.m.m.c, under L - H e n c e y C T-* C E~.
4.4. LEMMA. Let T and 7" be Frdchet equivalent continuous mappings from
Q an 0 respectively into $3. Then
K(x, T, Q ) = K ( x , T, Q) for x E S 3.
Proof. This follows directly from the definition o[ K(x, T, Q), K(x, T, Q)
and the lemma in w 4.3.
4.5. THEOREM. Let T be a continuous mappings as given in w 3.1 Then
(1) A(T) =fK(x, T, Q) d/-/~.
Proof. Case 1. A ( T ) : ~ . (1) follows from the lemma in w 3.4.
Case 2. A ( T ) < c~. Then (see w 1.2) there is a continuous mapping ?"
from the unit square Q: 0 ~ u - ~ < 1, 0~<v-~< 1 into $3 such that 7" and T are
Fr~chet equivalent, .](u, v) (the Jacobian of T--) exists everywhere on Q except on a set of L2-measure zero and
A(T) : f ] ~ , v )dH, . O
By the theorem in w 3.9 and the lemma in w 4.4 we have
A(T)= aCr)=fK(x, T, Q)dm=fK(x, T, Q)dn . Columbus, Mars 1955
2 1 8 , ~ A a , S. M z C x , ~ S
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