Research Article On the Property N 1

Transcription

1 Abstract ad Applied Aalysis Volume 2016, Article ID , 5 pages Research Article O the Property N 1 StaisBaw Kowalczyk ad MaBgorzata Turowska Istitute of Mathematics, Pomeraia Uiversity i Słupsk, Ulica Arciszewskiego 22d, Słupsk, Polad Correspodece should be addressed to Małgorzata Turowska; malgorzata.turowska@apsl.edu.pl Received 2 November 2015; Accepted 7 February 2016 Academic Editor: Feliz Mihós Copyright 2016 S. Kowalczyk ad M. Turowska. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We costruct a cotiuous fuctio f:[0,1] R such that f possesses N 1 -property, but f does ot have approximate derivative o a set of full Lebesgue measure. This shows that Baach s Theorem cocerig differetiability of cotiuous fuctios with Lusi s property (N) does ot hold for N 1 -property. Some relevat properties are preseted. 1. Itroductio First we will specify some basic otatios. By E we deote the Lebesgue measure of E R.Forayf:I R,whereI is a iterval, by f Ewedeote the restrictio of f to E I ad the symbol f ap (x) stads for approximate derivative of f at x. Defiitio 1 (see [1]). Let D R be measurable. We say that f:d R has Lusi s property (N), if the image f(e) of every set E D of Lebesgue measure 0 has Lebesgue measure 0. This coditio was studied exhaustively; some of results ca be foud i [1]. For the preset paper the most importat is the followig. Theorem 2 (Third Baach Theorem, [1] Theorem 7.3). If f: [0, 1] R is cotiuous ad has Lusi s property (N),thef isdifferetiableoasetofpositivelebesguemeasure. I the preset paper we will study a similar property. Defiitio 3 (see [2, 3]). We say that f:d R, defied o a measurable set D R, hasn 1 -property, if the iverse image f 1 (E) of every set E RofLebesgue measure 0 has Lebesgue measure 0. Some of results cocerig N 1 -property are preseted i [2, 3]. I [2] a systematic study of N 1 -property for smooth ad almost everywhere differetiable fuctios ca be foud. Some applicatios of N 1 -property i fuctioal equatio ad geometric fuctio theory ca be foud i [4 6]. 2. Mai Results Our goal is to costruct a cotiuous fuctio f:[0,1] [0, 1] with N 1 -property which is ot approximately differetiable o a set of full measure. We start with the basic theorem. Theorem 4. Let B 1 = (2k 1)/2 : k 1,2,...,2 1 }, N}, B 2 =(2k 1)/2 +1/(3 2 ):k 1,2,...,2 1 }, N}, ad A = (0, 1) \ (B 1 B 2 ). There exists a homeomorphism f:a Asuch that (a1) f=f 1, (a2) f has Lusi s property (N) ad N 1 -property, (a3) f has o approximate derivative (fiite or ot) at ay x A. Proof. Let x=0.i 1 i 2 i deote a biary decompositio of x (0, 1). Itiseasilyseethat(2k 1)/2 +1/(3 2 ) = 0.i 1 i 2 i Therefore, x (0, 1) ad x=0.i 1 i 2 i belogs to A if ad oly if it has a biary decompositio x=0.i 1 i 2 i such that N :i 2 =0}=ℵ 0 = N :i 2 =1}or N :i 2 1 =0}=ℵ 0 = N :i 2 1 =1} (1)

2 2 Abstract ad Applied Aalysis (i other words x has ifiitely may 0s adifiitelymay 1sateveplacesorifiitelymay0sadifiitelymay1sat odd places). Let Δ k =(k/2,(k+1)/2 ) for k 0,1,...,2 1} ad N. Obviously,A 2 1 k=0 Δk for every N. Moreover, A Δ k = x A:x=0.i 1 i 2 i where Defie f:a Aby j=1 2 j i j =k } }. } (2) f (x) =0.i 1 i2 i 3i4 i 2 1i2, (3) where x=0.i 1 i 2 i ad i j =1 i j.iotherwords,f(x) = 0.m 1 m 2,wherem j =i j for odd j ad m j =1 i j for eve. By(1),f(x) A for x Aad f is well-defied. Moreover, directly from the defiitio of f, it follows that f is a bijectio ad the compositio f fis the idetity fuctio, whece f 1 =f. Moreover, by (2), for each N ad k 0,1,...,2 1}, k= =1 2 j i j, i j 0,1},wehave f (A Δ k ) =A Δk, (4) where k = =1 2 j m j. We claim that f is cotiuous. Fix x 0 A ad ε > 0. Choose 0 N such that 1/2 0 <ε. There exists k for which x 0 Δ k 0.By(4),f(A Δ k 0 )=A Δ k 0.Sice A Δ k 0 is a eighborhood of x 0 ad y 1 y 2 1/2 0 <εfor all y 1,y 2 Δ k 0,wecocludethatfis cotiuous at x 0.Thus, f is cotiuous, because x 0 was arbitrary. By the equality f= f 1, f is a homeomorphism. Now we will show coditio (a2). Let H Abe ay set of Lebesgue measure zero. Fix ay ε>0. There exists a ope i A set U Asuch that H Uad U < ε.let B = Δ k A:k 0,1,...,2 1}, N}. (5) Clearly, B is a base of the atural topology i A. Sice either ay two sets from B are disjoit or oe of them is cotaied i the other, it is easy to see that ay ope subset of A ca be represeted as a uio of some subfamily of pairwise disjoit sets from B. Thus,U= j J (Δ k j A),whereJ is at most coutable ad Δ k j 1 1 Δ k j 1 2 =0for j 1,j 2 J, j 1 =j 2.The, by (4), j J f(δ k j A)= j J (Δ k j A)is a ope i A set cotaiig f(h) ad j J Δk j A = Δk j A = U <ε. j J (6) Sice ε > 0 was arbitrary, f(h) = 0 ad f has Lusi s property (N). Sice f=f 1, f has also N 1 -property. Fially, we will show that f has o approximate derivative at ay x A.Fixx Aad a eve N. Thex Δ k for some k 2 1.Leti 1,i 2,...,i 0, 1} be such that k= =1 2 j i j. Moreover, let k be uderstood as before. It is clear that A Δ k =(A Δ4k ) (A Δ4k+1 ) f(a Δ 4k )=A Δ4k +1, f(a Δ 4k+1 )=A Δ4k, f(a Δ 4k+2 )=A Δ4k +3, f(a Δ 4k+3 +2 )=A Δ4k (remember that is eve). Note that f(y) f(x) y x if x Δ 4k Moreover, if x Δ 4k Δ k ad y A Δ4k+1 f(y) f(x) y x (A Δ 4k+2 ) (A Δ4k+3 ), (7) <0 (8) or x Δ4k+2 ad y A Δ4k+3. > = 1 3 ad y A Δ4k+2 Thus, if Nis eve ad x A Δ k Asuch that or x Δ4k+1 (9) ad y A Δ4k+3.,wecafidB, C B = C = 1 4 Δk, (10) f (x) f(y) <0 x y y B, (11) f (x) f(y) > 1 x y 3 y C. (12) Sicethisistrueforeveryeve ad Δ k = 1/2,we coclude that f has o approximate derivative (fiite or ot) at x.theproofiscompleted. From Baach s Theorem 2, we easily get the followig. Corollary 5. Ay fuctio f, defied o a iterval, which possesses Lusi s coditio (N) such that the set of discotiuity poits of f is fiite, is derivable at every poit of some set of positive Lebesgue measure. Meawhile, by Theorem 4, we have the followig. Theorem 6. There exists a bijectio g : [0, 1] [0, 1] such that (b1) g has Lusi s property (N) ad N 1 -property, (b2) the set of discotiuity poits of g is coutable, (b3) g has o approximate derivative at ay poit.

3 Abstract ad Applied Aalysis 3 Proof. Let B 1, B 2, A, adf be the same as i Theorem 4. It is easily see that every member of B 2 is of the form 0.i 1 i or 0.i 1 i for some N, except 1/6, 1/3, 2/3, 5/6. Defieφ : 0,1} B 1 B 2 0, 1} B 1 B 2 by φ (x) x for x 0,1, 1 6, 1 3, 2 3, 5 6 }, 0.j 1 j 2 j for x=0.i 1 i , = 0.j 1 j 2 j for x=0.i 1 i , 0.j 1 j 2 j 1 1 for x=0.i 1 i 1, (13) where j 2m 1 =i 2m 1 ad j 2m =1 i 2m.Itiseasytoseethatφ is a bijectio. Let g : [0, 1] [0, 1] be defied by g (x) = f (x) for x A, φ (x) for x 0, 1} B 1 B 2. (14) Fix x (0,1)\(B 1 B 2 ), x=0.i 1 i 2 i,adε>0.letm be a positive iteger such that 1/2 m <ε.theset C=0,1, 1 6, 1 3, 2 3, 5 6 } 2k 1 2 :k 1,2,...,2 1 }, m} 0.l 1 l : m} 0.l 1 l : m} (15) is fiite ad C 0,1} B 1 B 2. Hece, we ca fid δ (0,1/2 m+3 ) for which (x δ,x+δ) C = 0.Takeay y (x δ, x + δ) (B 1 B 2 ).Sice x y < 1/2 m+3 ad y B 1 B 2,wecocludey=0.i 1 i 2 i m or y = 0.i 1 i 2 i m or y = 0.i 1 i 2 i m 1. Hece, g(x) = f(x) = 0.j 1 j 2 j m ad g(y) = φ(y) = 0.j 1 j 2 j m or g(y) = φ(y) = 0.j 1 j 2 j m 1. Therefore, g(x) g(y) < 1/2 m < ε.sicef is cotiuous, g is cotiuous at x. Thus,wehaveprovedthatthesetofall discotiuity poits of g is cotaied i 0, 1} B 1 B 2. Therefore, g satisfies (b1), (b2), ad (b3). Theorem 7. For each ε (0, 1) there exist a closed owhere dese set F (0, 1) ad a homeomorphism h:f Fsuch that (c1) F > 1 ε, (c2) h=h 1, (c3) h has Lusi s property (N) ad N 1 -property, (c4) h has o approximate derivative (fiite or ot) at ay x F (more precisely, if h : [0, 1] [0, 1] is ay extesio of h the h has o approximate derivative (fiite or ot) at ay x F). Proof. Let B 1, B 2, A,adf be the same as i Theorem 4. Let x } =B 1 B 2.Fixε>0ad choose a sequece ( ) 0 of eve atural umbers satisfyig =02 1 j=+12 m j <ε, (16) < (17) For each 1there exists k 1,...,2 1}such that x ((k 1)/2,(k + 1)/2 ).Let B=(Δ 0 m 0 0} Δ 2m 0 1 m 0 1}) Sice Δ k 1 (Δ k 1 k 2 m } Δ k k 2 m } Δ k ). =( k 1 2 (18), k +1 2 m ), (19) B is a ope subset of [0, 1]. Moreover,B 1 B 2 0,1} B ad, by (16), B m m < ε 2. (20) By (4), i the proof of Theorem 4, for each 1there exist u, V 1,...,2 } such that f(δ k 1 A)=Δ u A, (21) f(δ k A)=Δ V A. Moreover, f(δ 0 m 0 A)=Δ i 0 A, f(δ 2m 0 1 m 0 A)=Δ i 1 A for some i 0,i 1 0,1,...,2 m 0 }.Hece, C=f(B A) =f(((δ 0 m 0 A) (Δ 2m 0 1 m 0 ((Δ k 1 A)) A) (Δ k A)))=(Δ i 0 A) (Δ i 1 A) ((Δ u A) (Δ V A)) =(Δ i 0 Δ i 1 (Δ u Δ V )) A. (22) (23) Agai, applyig (16), we have C = B < ε/2.moreover,sice [0, 1] \ A It B,theset B C=([0, 1] \A) B is ope i [0, 1]. (Δ i 0 Δ i 1 (Δ u Δ V )) (24)

4 4 Abstract ad Applied Aalysis Fially, put H = [0, 1] \ (B C).ItisclearthatH A, H is a closed subset of [0, 1],ad H > 1 2(ε/2) = 1 ε.sice f is a bijectio ad f=f 1,wehave f ((B C) A) =f(b A) f(c A) =C (B A) = (B C) A. (25) It follows that f(h) = H ad h=f His a homeomorphism. Fix x 0 Had N. There exists k 0,1,...,2 1} such that x 0 Δ k.certaily,δ k B C. Therefore, by (17), (B C) Δ k < (1/8) Δ k. By (10), (11), ad (12), x Δ k : f (x) f(x 0) > x x 0 <0} Δk, x Δ k : f (x) f(x 0) > 1 x x 0 3 } > Δk. (26) Therefore, ay extesio h : [0, 1] [0, 1] of h has o approximate derivative at x 0. Lemma 8. Let a, b, c, d R, a < b,adc < d.forevery ε (0, 1) thereexistaclosedowheredeseseth (a, b) ad a cotiuous ijectio g:h [c,d]such that (d1) H > (1/2)(b a), (d2) g 1 : g(h) H is cotiuous, (d3) g has Lusi s property (N) ad N 1 -property, (d4) if g : [a, b] [c, d] is ay extesio of g, the g has oapproximatederivative(fiiteorot)atayx H, (d5) g(mi H) c < ε, d g(max H) < ε,ad g(b ) g(a ) < ε for all N,where(a,b ): N} is the set of all coected compoets of (a,b)\h. Proof. Fix ε > 0 ad choose N such that 1/( + 1) < ε/2(d c). Leta=y 0 <x 1 <y 1 <x 2 < <x <y < x +1 =bbe a partitio of [a, b] such that y i x i = (1/( + 1))(b a) for i 1,...,}ad x j y j 1 = (1/( + 1) 2 )(b a) for j 1,...,, + 1}.Letψ : [a, b] [c, d] be a liear homeomorphism, ψ(x) = ((d c)/(b a))(x a) + c. By Theorem 7, there exist a closed owhere dese set F (0, 1) ad a homeomorphism h:f Fsatisfyig coditios (c2) (c4) such that F > ( + 1)/2.Foreachk 1,...,}defie liear homeomorphisms ψ k :[x k,y k ] [0,1], ψ k (x) = ad φ k : [0, 1] [ψ(x k ), ψ(y k )], 1 y k x k (x x k ), (27) φ k (x) =(ψ(y k ) ψ(x k )) x + ψ (x k ). (28) Moreover, let F k =ψ 1 k (F) for k. Obviously, each F k is a closed owhere dese subset of (x k,y k ). Besides, F k = F (y k x k ) = F (b a)/( + 1).Foreachk 1,...,}defie h k :F k [ψ(x k ), ψ(y k )] by h k =φ k h ψ k.itiseasytosee that each h k is a cotiuous ijectio, h k has Lusi s property (N) ad N 1 -property, ad, moreover, ay extesio of h k to [x k,y k ] is ot approximately differetiable at ay poit x F k.fially,leth= k=1 F k ad defie g:h [c,d]by g(x) = h k (x) for x F k, k 1,...,}. It is clear that H ad g satisfy coditios (d1) (d4). Let (α, β) be ay coected compoet of (a, b) \ H. If(α, β) [x k,y k ] for some k 1,...,}the g(β) g(α) ψ(y k) ψ(x k )= b a +1 d c b a = d c +1 <ε. If (α, β) [y k 1,x k ] for some k 2,...,}the (29) g(β) g(α) ψ(y k) ψ(x k 1 )=2 d c <ε. (30) +1 Similarly, g (mi F) c ψ(y 1) c= d c +1 + d c (+1) 2 = 2 (d c) (d c) < <ε. 2 (+1) +1 (31) Aalogously, d g(max F) < ε. Thiscompletestheproof. Now,wecaprovethemaitheoremofthepresetpaper. Theorem 9. There exists a cotiuous fuctio f:[0,1] [0, 1] such that f has N 1 -property, but f ap exists almost owhere. Proof. We will costruct iductively a sequece (F ) N of closed subsets of [0, 1] ad a sequece (f ) N of cotiuous fuctios f : [0, 1] [0, 1] such that (1) F F +1 ad F >1 1/2 for all 1, (2) f F k =f k F k for all >k, (3) f (x) f +1 (x) < 1/2 for 1,2,...} ad x [0, 1], (4) every f restricted to F has N 1 -property, (5) every extesio of f F has o approximate derivative at ay x F. First, we give a useful defiitio. If E (0, 1) is closed ad φ : E (0, 1), the by the liear extesio of f we mea ψ: [0, 1] [0, 1] such that ψ E=φ, ψ(0) = 0, ψ(1) = 1,ad ψ is liear o every closed iterval cotiguous to E 0,1}.It is clear that ψ is cotiuous if ad oly if φ is cotiuous. By Theorem 7, there exist a closed set F (0, 1), F > 1/2, adabijectiog 1 :F Fsatisfyig coditios (c1) (c4). Let F 1 = F ad f 1 : [0, 1] [0, 1] be the liear extesio of g 1.Thef 1 is cotiuous, f 1 has N 1 -property, ad every extesio of f 1 F 1 = g 1 has o approximate derivative at ay x F 1.

5 Abstract ad Applied Aalysis 5 Let ((a 1 k,b1 k )) k 1 bethefamilyofallcoectedcompoets of [0, 1] \ (F 1 0, 1}). Moreover, for every k N, let J 1 k be a ope iterval with edpoits f 1(a 1 k ) ad f 1(b 1 k ).By Lemma 8, for each k Nthere exist closed F 1 k (a1 k,b1 k ) ad g 1 k :F1 k J1 k satisfyig coditios (d1) (d5) with ε=1/2.let F 2 =F 1 k=1 F1 k ad let g 2 :F 2 F 1 k=1 J1 k be defied by g 2 (x) = f 1 (x) for x F 1 ad g 2 (x) = g 1 k (x) for x F1 k, k 1,2,...}.Weclaimthatg 2 is cotiuous. The cotiuity of g 2 at each poit of k=1 F1 k is obvious. Fix x 0 F 1 ad ε>0.ifx 0 isotisolatedfromtherightif 2, the there exist δ>0such that g 2 (x) g 2 (x 0 ) = g 1 (x) g 1 (x 0 ) < ε for x F 1 (x 0,x 0 +δ]ad F 2 (x 0,x 0 +δ)=(f 1 (x 0,x 0 + δ)) k K F 1 k for some K N.Sice g 2 (x) g 2 (x 0 ) (32) < max g 2 (a 1 k ) g 2 (x 0 ), g 2 (b 1 k ) g 2 (x 0 ) } for x J 1 k,wehave g 2(x) g 2 (x 0 ) < ε for x F 2 (x 0,x 0 +δ). Hece, g 2 is cotiuous from the right at x 0.Similarly,weca show that g 2 is cotiuous from the left at x 0.Sicex 0 was arbitrary, g 2 is cotiuous. Let f 2 be the liear extesio of g 2.ItisclearthatF 1 F 2, F 2 >1 1/4, f 2 F 1 =f 1 F 1, f 2 restricted to F 2 has N 1 -property, ad every extesio of f 2 F 2 =g 2 has o approximate derivative at ay x F 2.Moreover, f 2 (x) f 1 (x) < 1/2 for x [0,1]. Assume that closed sets F 1,...,F (0,1), F 1 F, ad cotiuous fuctios f r : [0, 1] [0, 1], r 1,...,}, are chose. Moreover, assume that for every r 2,...,}we have F r > 1 1/2 r, f r restricted to F r has N 1 -property, every extesio of f r F r has o approximate derivative at ay x F r, f r (x) f r 1 (x) < 1/2 r 1 for each x [0,1],ad f r F s =f s F s for every s 1,...,r 1}. Let ((a k,b k )) k 1 be the family of all coected compoets of [0, 1] \ (F 0,1}). Moreover, for every k Nlet J k be a ope iterval with edpoits f (a k ) ad f (b k ).By Lemma 8, for each k Nthere exist closed F k (a k,b k ) ad g k :F k J k satisfyig coditios (d1) (d5) with ε=1/2. Let F +1 =F k=1 F k ad let g +1 :F +1 F 1 k=1 J k be defied by g +1 (x) = f (x) for x F ad g +1 (x) = g k (x) for x F k, k 1,2,...}.Similarly,asithecaseofg 2,we ca check that g +1 is cotiuous. Let f +1 be the liear extesio of g +1.Itisclearthat F F +1, F +1 >1 1/2 +1, f +1 F =f F, f +1 restricted to F +1 has N 1 -property, ad every extesio of f +1 F +1 has o approximate derivative at ay x F +1. Moreover, f +1 (x) f (x) < 1/2 for x [0,1].Thus,we have proved iductively that there exist a sequece (F ) N of closed subsets of [0, 1] ad a sequece (f ) N of cotiuous fuctios f : [0, 1] [0, 1] satisfyig coditios (1) (5). Sice f +1 (x) f (x) < 1/2 for x [0,1] ad N, thesequece(f ) N is uiformly coverget to some cotiuous fuctio f : [0, 1] [0, 1]. Moreover,f F = f F for all N. Therefore, by (5), f has o approximate derivative at ay poit from F.Sice,by (1), F =1, f ap exists almost owhere. It remais to prove that f has N 1 -property. Take ay E [0, 1] of the Lebesgue measure zero. The, by (2), f 1 (E) = (F f 1 (E)) ([0, 1] \ F ) (F f 1 (E)) ([0, 1] \ F ). (33) Applyig (1) ad (4),wecocludethat f 1 (E) = 0.Thus,f has N 1 -property. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Refereces [1] S. Saks, Theory of the Itegral, Stechert, New York, NY, USA, 2d editio, [2] S. P. Poomarev, Submersios ad pre-images of sets of measure zero, Sibirskii Matematicheskii Zhural, vol.28,o.1, pp ,1987. [3] S. P. Poomarev, The N 1 -property of maps ad Luzi s coditio (N), Mathematical Notes,vol.58,o.3,pp , [4] M. Charalambides, O restrictig Cauchy-Pexider fuctioal equatios to submaifolds, Aequatioes Mathematicae, vol. 86, o.3,pp ,2013. [5] O.Martio,V.Ryazaov,U.Srebro,adE.Yakubov, Mappigs with fiite legth distortio, Joural d Aalyse Mathématique, vol. 93, pp , [6]R.R.SalimovadE.A.Sevost yaov, TheoryofrigQmappigs ad geometric fuctio theory, Matematicheski Sborik,vol.201,o.6,pp ,2010.

6 Advaces i Operatios Research Advaces i Decisio Scieces Applied Mathematics Algebra Probability ad Statistics The Scietific World Joural Iteratioal Differetial Equatios Submit your mauscripts at Iteratioal Advaces i Combiatorics Mathematical Physics Complex Aalysis Iteratioal Mathematics ad Mathematical Scieces Mathematical Problems i Egieerig Mathematics Discrete Mathematics Discrete Dyamics i Nature ad Society Fuctio Spaces Abstract ad Applied Aalysis Iteratioal Stochastic Aalysis Optimizatio