A NOTE ON INVARIANT MEASURES. Piotr Niemiec
|
|
- Elfrieda Willis
- 6 years ago
- Views:
Transcription
1 Opuscula Mathematica Vol. 31 No A NOTE ON INVARIANT MEASURES Piotr Niemiec Abstract. The aim of the paper is to show that if F is a family of cotiuous trasformatios of a oempty compact Hausdorff space Ω, the there is o F-ivariat probabilistic regular Borel measures o Ω iff there are ϕ 1,...,ϕ p F (for some p 2) ad a cotiuous fuctio u: lim if 1 R such that P σ S p u(x σ(1),...,x σ(p) ) = 0 ad P 1 (u Φk )(x 1,...,x p) 1 for each x 1,...,x p Ω, where Φ: (x 1,...,x p) (ϕ 1(x 1),...,ϕ p(x p)) ad Φ k is the k-th iterate of Φ. A modified versio of this result i case the family F geerates a equicotiuous semigroup is proved. Keywords: ivariat measures, equicotiuous semigroups, compact spaces. Mathematics Subject Classificatio: 28C10, 54H INTRODUCTION Ivariat measures are preset i may parts of mathematics, icludig harmoic aalysis, ergodic theory ad topological dyamics. Ergodic theory deals with a sigle measurable trasformatio which preserves a fixed measure ad it focuses o properties of the measure-theoretic discrete dyamical system obtaied i this way. The reader iterested i this subject is referred to stadard textbooks such as [6] or [4]. Aother approach to the aspect of ivariat measures, treated i the recet paper, is, i a sese, related to (commo) fixed poit theory ad it cocetrates o the problem of the existece of a measure preserved by all trasformatios of a fixed family. The most classical result i this topic is the Haar measure theorem which states that o every locally compact topological group there is a uique, up to a costat factor, positive regular Borel measure ivariat uder the left shifts of the group (see e.g. [5, 13] or [12]; for a much more geeral result see [9, 10, 19]). This meaigful result plays a importat role i abstract harmoic aalysis ad group represetatio theory ad gave foudatios to this ew brach of mathematics which is still widely ivestigated. This icludes ivariat measures for both the groups as well as the semigroups of cotiuous or measurable trasformatios actig o metric spaces, compact spaces or totally arbitrary topological spaces. There is a huge rage of literature cocerig 425
2 426 Piotr Niemiec this subject ad we metio here oly a part: [1,3,7,8,11,14 17,20] or a survey article [21] ad refereces therei. The recet paper deals with a arbitrary semigroup F of cotiuous trasformatios of a compact Hausdorff space. Our aim is to give a equivalet coditio for the existece of a Borel regular probabilistic measure ivariat uder each member of F. The coditio reduces the problem of the existece of the measure to the more friedly issue of the oexistece of a cotiuous fuctio satisfyig certai explicitly stated coditios. 2. PRELIMINARIES I this paper Ω is a oempty compact Hausdorff space ad S stads for the group of all permutatios of {1,...,}. Forσ S, let σ :Ω Ω be a fuctio defied by σ(x 1,...,x )=(x σ(1),...,x σ() ). Wheever Φ is a trasformatio of some set, Φ k deotes the k-th iterate of Φ. The algebra of all the cotiuous real-valued fuctios o Ω is deoted by C(Ω), B(Ω) deotes the σ-algebra of all the Borel subsets of Ω ad C(Ω, Ω) stads for the family of cotiuous trasformatios of Ω; M(Ω) is the vector space of all the (siged) real-valued regular Borel measures o Ω ad Prob(Ω) is its subset of probabilistic measures. The space M(Ω) is equipped with the stadard weak-* topology iduced by liear fuctioals of the form M(Ω) μ f dμ R, where f C(Ω). For Ω a cotiuous trasformatio ϕ: Ω Ω, let ˆϕ: M(Ω) M(Ω) be a trasformatio give by the formula ˆϕ(μ)(A) =μ(ϕ 1 (A)) (μ M(Ω), A B(Ω)). (Thus ˆϕ(μ) is the trasport of the measure μ uder the trasformatio ϕ.) The set Prob(Ω) is compact ad the trasformatio ˆϕ is cotiuous i the weak-* topology (for cotiuous ϕ). (For the proof of the secod statemet see e.g. [14].) For measures μ 1,...,μ M(Ω) ad trasformatios ϕ 1,...,ϕ :Ω Ω, we deote by μ 1... μ ( M(Ω )) ad ϕ 1... ϕ the product of μ 1,...,μ ad of ϕ 1,...,ϕ, respectively. Thus ϕ 1... ϕ :Ω Ω ad (ϕ 1... ϕ )(x 1,...,x )=(ϕ 1 (x 1 ),...,ϕ (x )). If F is a family of cotiuous trasformatios of Ω, we say that F is equicotiuous if ad oly if the closure of F i the compact-ope topology of C(Ω, Ω) is compact (i that topology). Equivaletly, F is equicotiuous if for ay poits x, y Ω ad every ope eighbourhood V Ω of the poit y there exist ope subsets U ad W of Ω such that x U, y W ad for each ϕ F, ϕ(u) V provided ϕ(x) W.IfF is equicotiuous ad h C(Ω), the the closure of the set h F = {h ϕ: ϕ F}i the orm topology of C(Ω) is compact. For proofs, details ad more iformatio o the compact-ope topology the reader is referred to [2]. For a family F C(Ω, Ω), let Iv(F) Prob(Ω) be the set of all the F-ivariat measures, i.e. a measure μ Prob(Ω) belogs to Iv(F) if ad oly if ˆϕ(μ) =μ for each ϕ F. Note, for example, that Iv( ) = Prob(Ω). IfF = {ϕ 1,...,ϕ }, we shall write Iv(ϕ 1,...,ϕ ) istead of Iv({ϕ 1,...,ϕ }). The set Iv(F) is always compact (i the weak-* topology) ad the followig result has etered folklore i ergodic theory:
3 A ote o ivariat measures 427 Theorem 2.1. If ϕ C(Ω, Ω) ad μ Prob(Ω), the every limit poit of the sequece ( 1 1 (ˆϕ)k (μ)) =1 belogs to Iv(ϕ). A variatio of the above result is the crucial key i the proof of Markov s-kakutai s fixed poit theorem see e.g. [18]. Theorem 2.1 is i fact a special case of this variatio. For simplicity, let A p (Ω) (where p 2) deote the family of all cotiuous fuctios u: R such that u σ 0. (2.1) σ S p The followig result is well kow ad easy to prove. Lemma 2.2. For a family F C(Ω, Ω), the followig coditios are equivalet: (i) the set Iv(F) is empty, (ii) there are a atural umber N 2 ad ϕ 1,...,ϕ N F such that Iv(ϕ 1,...,ϕ N )=. 3. MAIN RESULTS Lemma 2.2 says that we may restrict our ivestigatios to fiite sets of trasformatios, which shall be doe i the sequel. For simplicity, we fix the situatio. Let ϕ 1,...,ϕ p (p 2) be members of C(Ω, Ω) ad Φ=ϕ 1... ϕ p : (ote that Iv(Φ) M( )). Additioally, let Iv(S p ) be the collectio of all siged real-valued regular Borel measures o, ivariat uder all permutatios of variables. The followig simple result may be iterestig i itself. Lemma 3.1. Iv(ϕ 1,...,ϕ p ) is oempty iff Iv(Φ) Iv(S p ). Proof. It is easy to check that if μ Iv(ϕ 1,...,ϕ p ), the λ Iv(Φ) Iv(S p ) for λ = μ... μ M( ). Coversely, if λ belogs to both the sets Iv(Φ) ad Iv(S p ), the it is easily verified that a measure μ M(Ω) defied by μ(a) =λ(a 1 )(A B(Ω)) is ϕ j -ivariat for j =1,...,p. So, if we wat to kow whe Iv(ϕ 1,...,ϕ p )=, it is eough to verify whe the sets Iv(Φ) ad Iv(S p ) are disjoit. Sice the first of them is covex ad compact ad the latter is a closed (i the weak-* topology) liear subspace of M( ),thus by the separatio theorem they are disjoit if ad oly if there is u C( ) such that u dμ =0for μ Iv(S p ), but for some positive t we have u dλ t (3.1) for ay λ Iv(Φ).
4 428 Piotr Niemiec The proof of the followig fact is immediate. Lemma 3.2. Let u C( ). The u dμ =0for each μ Iv(S p ) if ad oly if u A p (Ω). The property (3.1) of the fuctio u which separates the sets Iv(S p ) ad Iv(Φ) ca be reformulated as follows: Lemma 3.3. For a fuctio u C( ) ad a umber t R the followig coditios are equivalet: (i) the iequality (3.1) holds for every λ Iv(Φ), 1 1 (ii) lim if (u Φk )(z) t for each z. Proof. The implicatio (ii) = (i) follows from Fatou s lemma. Ideed, take m R such that u(z) m for each z. The for λ Iv(Φ) we obtai: ( 1 1 lim if ) u Φ k m dλ ( 1 1 lim if (t m) dλ = t m, ) u Φ k m dλ (3.2) but ( 1 1 ) u Φ k m dλ = u Φ k 1 dλ m = u dˆφ k (λ) m = = 1 1 u dλ m = u dλ m. For the coverse implicatio, fix z ad take a subsequece (s k ) k of the sequece s = 1 1 u(φj (z)) such that lim s k = lim if s. k Let δ be the Dirac measure o with the atom at z. Sice Prob( ) is compact, the sequece μ k = 1 k 1 k ˆΦ j (δ) has a limit poit i Prob( ), say λ. By Theorem 2.1, λ Iv(Φ) ad hece u dλ t. Fially, sice the fuctio M( ) ν u dν R is cotiuous, therefore the itegral u dλ is a limit poit of the sequece u dμ k. But u dμ k = s k lim if s (k + ), which fiishes the proof.
5 A ote o ivariat measures 429 Now puttig together the above facts, we obtai the mai result of the paper. Theorem 3.4. The followig coditios are equivalet: (i) Iv(ϕ 1,...,ϕ p )=, (ii) there is u A p (Ω) such that for each z, lim if 1 1 (u Φ k )(z) 1. (3.3) Before we stregthe coditio (ii) of the foregoig theorem i case the family {ϕ 1,...,ϕ p } geerates a equicotiuous semigroup, we shall prove the followig Lemma 3.5. Let v A p (Ω). The followig coditios are equivalet: (i) there are c>0 ad a umber 0 1 such that 1 c(v Φk )(z) >for each z ad 0, (ii) there is m 0 such that the fuctio m v Φk has oly positive values. Each of the above coditios implies that Iv(ϕ 1,...,ϕ p )=. Proof. Thaks to Theorem 3.4, it is eough to prove the equivalece of (i) ad (ii). m To see that (ii) implies (i), put ε = if z (v Φk )(z). By (ii) ad the compactess of Ω, ε>0. For simplicity, put l = m +1 ( 1) ad c = 2l ε > 0. Let t = if { k (v Φj )(z): k {0,...,l 1}, z } (observe that t 0, because v is either costatly equal 0 or is ot oegative). Fially, take 0 1 such that t ε > 1 c (3.4) for every 0. Let be a arbitrary atural umber o less tha 0. Express i the form = sl + r, where s 1 ad 0 r<l. From the defiitio of ε it follows that l 1 (v Φj )(Φ ql+r (z)) ε for every z ad q =0,...,s 1. Furthermore, r 1 (v Φj )(z) t (this is true also for r =0, uder the agreemet that =0). Hece, by (3.4): 1 r 1 s 1 l 1 c(v Φ j )(z) =c (v Φ j )(z)+c (v Φ ql+r+j )(z) ( c(t + sε) =c ( 2 >c c ) =, c q=0 t + r l ) ( ε >c t + l l ) ( ε ) ε = c l + t ε > which fiishes the proof of the implicatio (ii) = (i). The coverse implicatio is immediate.
6 430 Piotr Niemiec The aouced stregtheig of Theorem 3.4 has the followig form: Theorem 3.6. If the family {ϕ 1,...,ϕ p } geerates a equicotiuous semigroup (with the actio of compositio), the the followig coditios are equivalet: (i) Iv(ϕ 1,...,ϕ p )=, (ii) there are v A p (Ω) ad a umber 0 1 such that 1 (v Φk )(z) >for each 0 ad z. Proof. It suffices to prove that (i) implies (ii). Suppose that the set Iv(ϕ 1,...,ϕ p ) is empty ad let u A p (Ω) be as i the statemet of the coditio (ii) of Theorem 3.4. Put v =2u A p (Ω). We shall show that v is the fuctio which we are lookig for. Suppose, to the cotrary, there exist a icreasig sequece ( k ) k of atural umbers ad a sequece (z k ) k of elemets of such that (v Φ j )(z k ) k (k 1). (3.5) k 1 Let v k = 1 k 1 k v Φj. Sice the semigroup geerated by F is equicotiuous, hece so is the semigroup {Φ j : j 0}. This implies that the uiform closure of V = {v Φ j : j 0} is compact ad therefore the closed covex hull of V i C( ) is compact as well. So, replacig evetually the sequece (v k ) k by a suitable subsequece, we may assume that (v k ) k is uiformly coverget to some v 0 C( ).Letz 0 be a limit poit of the sequece (z k ) k. From the uiform covergece of (v k ) k to v 0 we ifer that v 0 (z 0 ) is a limit poit of the sequece (v k (z k )) k ad thus, by (3.5), v 0 (z 0 ) 1. But o the other had, by (3.3): 1 v 0 (z 0 ) = lim v 1 k(z 0 ) lim if k which is a cotradictio. (v Φ j )(z 0 ) = 2 lim if 1 1 (u Φ j )(z 0 ) 2, Other coditios for the oemptiess of Iv(F) i case the family F geerates a equicotiuous semigroup ca be foud i [14]. REFERENCES [1] R.B. Chuaqui, Measures ivariat uder a group of trasformatios, Pacific J. Math. 68 (1977), [2] R. Egelkig, Geeral Topology, PWN Polish Scietific Publishers, Warszawa, [3] P. Erdös, R.D. Mauldi, The oexistece of certai ivariat measures, Proc. Amer. Math. Soc. 59 (1976), [4] N.A. Friedma, Itroductio to Ergodic Theory, Va Nostrad Reihold Compay, 1970.
7 A ote o ivariat measures 431 [5] P.R. Halmos, Measure Theory, Va Nostrad, New York, [6] P.R. Halmos, Lectures o Ergodic Theory, Publ. Math. Soc. Japa, Tokyo, [7] V. Kaa, S.R. Raju, The oexistece of ivariat uiversal measures o semigroups, Proc. Amer. Math. Soc. 78 (1980), [8] A.B. Khararazishvili, Trasformatio groups ad ivariat measures. Set-theoretic aspects, World Scietific Publishig Co., Ic., River Edge, NJ, [9] L.H. Loomis, Abstract cogruece ad the uiqueess of Haar measure, A. Math. 46 (1945), [10] L.H. Loomis, Haar measure i uiform structures, Duke Math. J. 16 (1949), [11] J. Mycielski, Remarks o ivariat measures i metric spaces, Coll. Math. 32 (1974), [12] L. Nachbi, The Haar Itegral, D. Va Nostrad Compay, Ic., Priceto-New Jersey-Toroto-New York-Lodo, [13] J. vo Neuma, Zum Haarsche Mass i topologische Gruppe, Compos. Math. 1 (1934), [14] P. Niemiec, Ivariat measures for equicotiuous semigroups of cotiuous trasformatios of a compact Hausdorff space, Topology Appl. 153 (2006), [15] A. Pelc, Semiregular ivariat measures o abelia groups, Proc. Amer. Math. Soc. 86 (1982), [16] D. Ramachadra, M. Misiurewicz, Hopf s theorem o ivariat measures for a group of trasformatios, Studia Math. 74 (1982), [17] J.M. Roseblatt, Equivalet ivariat measures, Israel J. Math. 17 (1974), [18] W. Rudi, Fuctioal Aalysis, McGraw-Hill Sciece, [19] R.C. Steilage, O Haar measure i locally compact T 2 spaces, Amer. J. Math. 97 (1975), [20] P. Zakrzewski, The existece of ivariat σ-fiite measures for a group of trasformatios, Israel J. Math. 83 (1993), [21] P. Zakrzewski, Measures o Algebraic-Topological Structures, Hadbook of Measure Thoery, E. Pap, ed., Elsevier, Amsterdam, 2002, Piotr Niemiec piotr.iemiec@uj.edu.pl Jagielloia Uiversity Istitute of Mathematics ul. Łojasiewicza 6, Kraków, Polad Received: July 30, Revised: October 29, Accepted: November 11, 2010.
Product measures, Tonelli s and Fubini s theorems For use in MAT3400/4400, autumn 2014 Nadia S. Larsen. Version of 13 October 2014.
Product measures, Toelli s ad Fubii s theorems For use i MAT3400/4400, autum 2014 Nadia S. Larse Versio of 13 October 2014. 1. Costructio of the product measure The purpose of these otes is to preset the
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More informationLecture Notes for Analysis Class
Lecture Notes for Aalysis Class Topological Spaces A topology for a set X is a collectio T of subsets of X such that: (a) X ad the empty set are i T (b) Uios of elemets of T are i T (c) Fiite itersectios
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationLecture 3 The Lebesgue Integral
Lecture 3: The Lebesgue Itegral 1 of 14 Course: Theory of Probability I Term: Fall 2013 Istructor: Gorda Zitkovic Lecture 3 The Lebesgue Itegral The costructio of the itegral Uless expressly specified
More informationA NOTE ON SPECTRAL CONTINUITY. In Ho Jeon and In Hyoun Kim
Korea J. Math. 23 (2015), No. 4, pp. 601 605 http://dx.doi.org/10.11568/kjm.2015.23.4.601 A NOTE ON SPECTRAL CONTINUITY I Ho Jeo ad I Hyou Kim Abstract. I the preset ote, provided T L (H ) is biquasitriagular
More informationChapter 7 Isoperimetric problem
Chapter 7 Isoperimetric problem Recall that the isoperimetric problem (see the itroductio its coectio with ido s proble) is oe of the most classical problem of a shape optimizatio. It ca be formulated
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationThe Wasserstein distances
The Wasserstei distaces March 20, 2011 This documet presets the proof of the mai results we proved o Wasserstei distaces themselves (ad ot o curves i the Wasserstei space). I particular, triagle iequality
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationA FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE. Abdolrahman Razani (Received September 2004)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 35 (2006), 109 114 A FIXED POINT THEOREM IN THE MENGER PROBABILISTIC METRIC SPACE Abdolrahma Razai (Received September 2004) Abstract. I this article, a fixed
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationBoundaries and the James theorem
Boudaries ad the James theorem L. Vesely 1. Itroductio The followig theorem is importat ad well kow. All spaces cosidered here are real ormed or Baach spaces. Give a ormed space X, we deote by B X ad S
More informationf n (x) f m (x) < ɛ/3 for all x A. By continuity of f n and f m we can find δ > 0 such that d(x, x 0 ) < δ implies that
Lecture 15 We have see that a sequece of cotiuous fuctios which is uiformly coverget produces a limit fuctio which is also cotiuous. We shall stregthe this result ow. Theorem 1 Let f : X R or (C) be a
More informationA NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS
Acta Math. Hugar., 2007 DOI: 10.1007/s10474-007-7013-6 A NOTE ON INVARIANT SETS OF ITERATED FUNCTION SYSTEMS L. L. STACHÓ ad L. I. SZABÓ Bolyai Istitute, Uiversity of Szeged, Aradi vértaúk tere 1, H-6720
More informationRiesz-Fischer Sequences and Lower Frame Bounds
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 1 (00), No., 305 314 Riesz-Fischer Sequeces ad Lower Frame Bouds P. Casazza, O. Christese, S. Li ad A. Lider Abstract.
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationLecture 3 : Random variables and their distributions
Lecture 3 : Radom variables ad their distributios 3.1 Radom variables Let (Ω, F) ad (S, S) be two measurable spaces. A map X : Ω S is measurable or a radom variable (deoted r.v.) if X 1 (A) {ω : X(ω) A}
More informationFUNDAMENTALS OF REAL ANALYSIS by
FUNDAMENTALS OF REAL ANALYSIS by Doğa Çömez Backgroud: All of Math 450/1 material. Namely: basic set theory, relatios ad PMI, structure of N, Z, Q ad R, basic properties of (cotiuous ad differetiable)
More informationON THE EXTENDED AND ALLAN SPECTRA AND TOPOLOGICAL RADII. Hugo Arizmendi-Peimbert, Angel Carrillo-Hoyo, and Jairo Roa-Fajardo
Opuscula Mathematica Vol. 32 No. 2 2012 http://dx.doi.org/10.7494/opmath.2012.32.2.227 ON THE EXTENDED AND ALLAN SPECTRA AND TOPOLOGICAL RADII Hugo Arizmedi-Peimbert, Agel Carrillo-Hoyo, ad Jairo Roa-Fajardo
More informationMcGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz fuctios. Let Lip K be the set of all fuctios cotiuous fuctios o [, 1] satisfyig a Lipschitz
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationProperties of Fuzzy Length on Fuzzy Set
Ope Access Library Joural 206, Volume 3, e3068 ISSN Olie: 2333-972 ISSN Prit: 2333-9705 Properties of Fuzzy Legth o Fuzzy Set Jehad R Kider, Jaafar Imra Mousa Departmet of Mathematics ad Computer Applicatios,
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 21 11/27/2013 Fuctioal Law of Large Numbers. Costructio of the Wieer Measure Cotet. 1. Additioal techical results o weak covergece
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationSolution. 1 Solutions of Homework 1. Sangchul Lee. October 27, Problem 1.1
Solutio Sagchul Lee October 7, 017 1 Solutios of Homework 1 Problem 1.1 Let Ω,F,P) be a probability space. Show that if {A : N} F such that A := lim A exists, the PA) = lim PA ). Proof. Usig the cotiuity
More informationReal Numbers R ) - LUB(B) may or may not belong to B. (Ex; B= { y: y = 1 x, - Note that A B LUB( A) LUB( B)
Real Numbers The least upper boud - Let B be ay subset of R B is bouded above if there is a k R such that x k for all x B - A real umber, k R is a uique least upper boud of B, ie k = LUB(B), if () k is
More informationABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS
ABOUT CHAOS AND SENSITIVITY IN TOPOLOGICAL DYNAMICS EDUARD KONTOROVICH Abstract. I this work we uify ad geeralize some results about chaos ad sesitivity. Date: March 1, 005. 1 1. Symbolic Dyamics Defiitio
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationSingular Continuous Measures by Michael Pejic 5/14/10
Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable
More informationAxioms of Measure Theory
MATH 532 Axioms of Measure Theory Dr. Neal, WKU I. The Space Throughout the course, we shall let X deote a geeric o-empty set. I geeral, we shall ot assume that ay algebraic structure exists o X so that
More informationMi-Hwa Ko and Tae-Sung Kim
J. Korea Math. Soc. 42 2005), No. 5, pp. 949 957 ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF NEGATIVELY ORTHANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko ad Tae-Sug Kim Abstract. For weighted sum of a sequece
More informationDANIELL AND RIEMANN INTEGRABILITY
DANIELL AND RIEMANN INTEGRABILITY ILEANA BUCUR We itroduce the otio of Riema itegrable fuctio with respect to a Daiell itegral ad prove the approximatio theorem of such fuctios by a mootoe sequece of Jorda
More informationThe Borel hierarchy classifies subsets of the reals by their topological complexity. Another approach is to classify them by size.
Lecture 7: Measure ad Category The Borel hierarchy classifies subsets of the reals by their topological complexity. Aother approach is to classify them by size. Filters ad Ideals The most commo measure
More informationb i u x i U a i j u x i u x j
M ath 5 2 7 Fall 2 0 0 9 L ecture 1 9 N ov. 1 6, 2 0 0 9 ) S ecod- Order Elliptic Equatios: Weak S olutios 1. Defiitios. I this ad the followig two lectures we will study the boudary value problem Here
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationAN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS. Robert DEVILLE and Catherine FINET
2001 vol. XIV, um. 1, 95-104 ISSN 1139-1138 AN EXTENSION OF SIMONS INEQUALITY AND APPLICATIONS Robert DEVILLE ad Catherie FINET Abstract This article is devoted to a extesio of Simos iequality. As a cosequece,
More informationIntroduction to Optimization Techniques
Itroductio to Optimizatio Techiques Basic Cocepts of Aalysis - Real Aalysis, Fuctioal Aalysis 1 Basic Cocepts of Aalysis Liear Vector Spaces Defiitio: A vector space X is a set of elemets called vectors
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationApproximation by Superpositions of a Sigmoidal Function
Zeitschrift für Aalysis ud ihre Aweduge Joural for Aalysis ad its Applicatios Volume 22 (2003, No. 2, 463 470 Approximatio by Superpositios of a Sigmoidal Fuctio G. Lewicki ad G. Mario Abstract. We geeralize
More informationReal and Complex Analysis, 3rd Edition, W.Rudin
Real ad Complex Aalysis, 3rd ditio, W.Rudi Chapter 6 Complex Measures Yug-Hsiag Huag 206/08/22. Let ν be a complex measure o (X, M ). If M, defie { } µ () = sup ν( j ) : N,, 2, disjoit, = j { } ν () =
More informationThe Australian Journal of Mathematical Analysis and Applications
The Australia Joural of Mathematical Aalysis ad Applicatios Volume 6, Issue 1, Article 10, pp. 1-10, 2009 DIFFERENTIABILITY OF DISTANCE FUNCTIONS IN p-normed SPACES M.S. MOSLEHIAN, A. NIKNAM AND S. SHADKAM
More informationBIRKHOFF ERGODIC THEOREM
BIRKHOFF ERGODIC THEOREM Abstract. We will give a proof of the poitwise ergodic theorem, which was first proved by Birkhoff. May improvemets have bee made sice Birkhoff s orgial proof. The versio we give
More informationOn Weak and Strong Convergence Theorems for a Finite Family of Nonself I-asymptotically Nonexpansive Mappings
Mathematica Moravica Vol. 19-2 2015, 49 64 O Weak ad Strog Covergece Theorems for a Fiite Family of Noself I-asymptotically Noexpasive Mappigs Birol Güdüz ad Sezgi Akbulut Abstract. We prove the weak ad
More informationSOME GENERALIZATIONS OF OLIVIER S THEOREM
SOME GENERALIZATIONS OF OLIVIER S THEOREM Alai Faisat, Sait-Étiee, Georges Grekos, Sait-Étiee, Ladislav Mišík Ostrava (Received Jauary 27, 2006) Abstract. Let a be a coverget series of positive real umbers.
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationCharacter rigidity for lattices and commensurators I after Creutz-Peterson
Character rigidity for lattices ad commesurators I after Creutz-Peterso Talk C3 for the Arbeitsgemeischaft o Superridigity held i MFO Oberwolfach, 31st March - 4th April 2014 1 Sve Raum 1 Itroductio The
More informationOn equivalent strictly G-convex renormings of Banach spaces
Cet. Eur. J. Math. 8(5) 200 87-877 DOI: 0.2478/s533-00-0050-3 Cetral Europea Joural of Mathematics O equivalet strictly G-covex reormigs of Baach spaces Research Article Nataliia V. Boyko Departmet of
More informationThe random version of Dvoretzky s theorem in l n
The radom versio of Dvoretzky s theorem i l Gideo Schechtma Abstract We show that with high probability a sectio of the l ball of dimesio k cε log c > 0 a uiversal costat) is ε close to a multiple of the
More informationLaw of the sum of Bernoulli random variables
Law of the sum of Beroulli radom variables Nicolas Chevallier Uiversité de Haute Alsace, 4, rue des frères Lumière 68093 Mulhouse icolas.chevallier@uha.fr December 006 Abstract Let be the set of all possible
More informationAn alternative proof of a theorem of Aldous. concerning convergence in distribution for martingales.
A alterative proof of a theorem of Aldous cocerig covergece i distributio for martigales. Maurizio Pratelli Dipartimeto di Matematica, Uiversità di Pisa. Via Buoarroti 2. I-56127 Pisa, Italy e-mail: pratelli@dm.uipi.it
More informationOn the Variations of Some Well Known Fixed Point Theorem in Metric Spaces
Turkish Joural of Aalysis ad Number Theory, 205, Vol 3, No 2, 70-74 Available olie at http://pubssciepubcom/tjat/3/2/7 Sciece ad Educatio Publishig DOI:0269/tjat-3-2-7 O the Variatios of Some Well Kow
More informationThe Pointwise Ergodic Theorem and its Applications
The Poitwise Ergodic Theorem ad its Applicatios Itroductio Peter Oberly 11/9/2018 Algebra has homomorphisms ad topology has cotiuous maps; i these otes we explore the structure preservig maps for measure
More informationA Fixed Point Result Using a Function of 5-Variables
Joural of Physical Scieces, Vol., 2007, 57-6 Fixed Poit Result Usig a Fuctio of 5-Variables P. N. Dutta ad Biayak S. Choudhury Departmet of Mathematics Begal Egieerig ad Sciece Uiversity, Shibpur P.O.:
More informationFinal Solutions. 1. (25pts) Define the following terms. Be as precise as you can.
Mathematics H104 A. Ogus Fall, 004 Fial Solutios 1. (5ts) Defie the followig terms. Be as recise as you ca. (a) (3ts) A ucoutable set. A ucoutable set is a set which ca ot be ut ito bijectio with a fiite
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationON THE FUZZY METRIC SPACES
The Joural of Mathematics ad Computer Sciece Available olie at http://www.tjmcs.com The Joural of Mathematics ad Computer Sciece Vol.2 No.3 2) 475-482 ON THE FUZZY METRIC SPACES Received: July 2, Revised:
More informationON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX0000-0 ON MEAN ERGODIC CONVERGENCE IN THE CALKIN ALGEBRAS MARCH T. BOEDIHARDJO AND WILLIAM B. JOHNSON 2
More informationTheorem 3. A subset S of a topological space X is compact if and only if every open cover of S by open sets in X has a finite subcover.
Compactess Defiitio 1. A cover or a coverig of a topological space X is a family C of subsets of X whose uio is X. A subcover of a cover C is a subfamily of C which is a cover of X. A ope cover of X is
More informationCouncil for Innovative Research
ABSTRACT ON ABEL CONVERGENT SERIES OF FUNCTIONS ERDAL GÜL AND MEHMET ALBAYRAK Yildiz Techical Uiversity, Departmet of Mathematics, 34210 Eseler, Istabul egul34@gmail.com mehmetalbayrak12@gmail.com I this
More informationEquivalent Banach Operator Ideal Norms 1
It. Joural of Math. Aalysis, Vol. 6, 2012, o. 1, 19-27 Equivalet Baach Operator Ideal Norms 1 Musudi Sammy Chuka Uiversity College P.O. Box 109-60400, Keya sammusudi@yahoo.com Shem Aywa Maside Muliro Uiversity
More informationSeveral properties of new ellipsoids
Appl. Math. Mech. -Egl. Ed. 008 9(7):967 973 DOI 10.1007/s10483-008-0716-y c Shaghai Uiversity ad Spriger-Verlag 008 Applied Mathematics ad Mechaics (Eglish Editio) Several properties of ew ellipsoids
More informationLecture 19: Convergence
Lecture 19: Covergece Asymptotic approach I statistical aalysis or iferece, a key to the success of fidig a good procedure is beig able to fid some momets ad/or distributios of various statistics. I may
More informationVECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS
Dedicated to Professor Philippe G. Ciarlet o his 70th birthday VECTOR SEMINORMS, SPACES WITH VECTOR NORM, AND REGULAR OPERATORS ROMULUS CRISTESCU The rst sectio of this paper deals with the properties
More informationSh. Al-sharif - R. Khalil
Red. Sem. Mat. Uiv. Pol. Torio - Vol. 62, 2 (24) Sh. Al-sharif - R. Khalil C -SEMIGROUP AND OPERATOR IDEALS Abstract. Let T (t), t
More informationSome vector-valued statistical convergent sequence spaces
Malaya J. Mat. 32)205) 6 67 Some vector-valued statistical coverget sequece spaces Kuldip Raj a, ad Suruchi Padoh b a School of Mathematics, Shri Mata Vaisho Devi Uiversity, Katra-82320, J&K, Idia. b School
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationChapter 0. Review of set theory. 0.1 Sets
Chapter 0 Review of set theory Set theory plays a cetral role i the theory of probability. Thus, we will ope this course with a quick review of those otios of set theory which will be used repeatedly.
More informationCHAPTER 5 SOME MINIMAX AND SADDLE POINT THEOREMS
CHAPTR 5 SOM MINIMA AND SADDL POINT THORMS 5. INTRODUCTION Fied poit theorems provide importat tools i game theory which are used to prove the equilibrium ad eistece theorems. For istace, the fied poit
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More informationA Note on the Kolmogorov-Feller Weak Law of Large Numbers
Joural of Mathematical Research with Applicatios Mar., 015, Vol. 35, No., pp. 3 8 DOI:10.3770/j.iss:095-651.015.0.013 Http://jmre.dlut.edu.c A Note o the Kolmogorov-Feller Weak Law of Large Numbers Yachu
More informationOn Strictly Point T -asymmetric Continua
Volume 35, 2010 Pages 91 96 http://topology.aubur.edu/tp/ O Strictly Poit T -asymmetric Cotiua by Leobardo Ferádez Electroically published o Jue 19, 2009 Topology Proceedigs Web: http://topology.aubur.edu/tp/
More informationA REMARK ON A PROBLEM OF KLEE
C O L L O Q U I U M M A T H E M A T I C U M VOL. 71 1996 NO. 1 A REMARK ON A PROBLEM OF KLEE BY N. J. K A L T O N (COLUMBIA, MISSOURI) AND N. T. P E C K (URBANA, ILLINOIS) This paper treats a property
More informationA NOTE ON LEBESGUE SPACES
Volume 6, 1981 Pages 363 369 http://topology.aubur.edu/tp/ A NOTE ON LEBESGUE SPACES by Sam B. Nadler, Jr. ad Thelma West Topology Proceedigs Web: http://topology.aubur.edu/tp/ Mail: Topology Proceedigs
More informationSome p-adic congruences for p q -Catalan numbers
Some p-adic cogrueces for p q -Catala umbers Floria Luca Istituto de Matemáticas Uiversidad Nacioal Autóoma de México C.P. 58089, Morelia, Michoacá, México fluca@matmor.uam.mx Paul Thomas Youg Departmet
More informationOn scaling entropy sequence of dynamical system
O scalig etropy sequece of dyamical system P. B. Zatitskiy arxiv:1409.6134v1 [math.ds] 22 Sep 2014 November 6, 2017 Abstract We preset a series of statemets about scalig etropy, a metric ivariat of dyamical
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationStrong Convergence Theorems According. to a New Iterative Scheme with Errors for. Mapping Nonself I-Asymptotically. Quasi-Nonexpansive Types
It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics
More informationMathematica Slovaca. Detlef Plachky A version of the strong law of large numbers universal under mappings
Mathematica Slovaca Detlef Plachky A versio of the strog law of large umbers uiversal uder mappigs Mathematica Slovaca, Vol. 49 (1999), No. 2, 229--233 Persistet URL: http://dml.cz/dmlcz/131004 Terms of
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationApplication to Random Graphs
A Applicatio to Radom Graphs Brachig processes have a umber of iterestig ad importat applicatios. We shall cosider oe of the most famous of them, the Erdős-Réyi radom graph theory. 1 Defiitio A.1. Let
More informationFixed Point Theorems for Expansive Mappings in G-metric Spaces
Turkish Joural of Aalysis ad Number Theory, 7, Vol. 5, No., 57-6 Available olie at http://pubs.sciepub.com/tjat/5//3 Sciece ad Educatio Publishig DOI:.69/tjat-5--3 Fixed Poit Theorems for Expasive Mappigs
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationChapter IV Integration Theory
Chapter IV Itegratio Theory Lectures 32-33 1. Costructio of the itegral I this sectio we costruct the abstract itegral. As a matter of termiology, we defie a measure space as beig a triple (, A, µ), where
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationStability of a Monomial Functional Equation on a Restricted Domain
mathematics Article Stability of a Moomial Fuctioal Equatio o a Restricted Domai Yag-Hi Lee Departmet of Mathematics Educatio, Gogju Natioal Uiversity of Educatio, Gogju 32553, Korea; yaghi2@hamail.et
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationSome Tauberian theorems for weighted means of bounded double sequences
A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. Some Tauberia theorems for weighted meas of bouded double sequeces Cemal Bele Received: 22.XII.202 / Revised: 24.VII.203/ Accepted: 3.VII.203
More informationA 2nTH ORDER LINEAR DIFFERENCE EQUATION
A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy
More informationSome Common Fixed Point Theorems in Cone Rectangular Metric Space under T Kannan and T Reich Contractive Conditions
ISSN(Olie): 319-8753 ISSN (Prit): 347-671 Iteratioal Joural of Iovative Research i Sciece, Egieerig ad Techology (A ISO 397: 7 Certified Orgaizatio) Some Commo Fixed Poit Theorems i Coe Rectagular Metric
More informationCARLEMAN INTEGRAL OPERATORS AS MULTIPLICATION OPERATORS AND PERTURBATION THEORY
Kragujevac Joural of Mathematics Volume 41(1) (2017), Pages 71 80. CARLEMAN INTEGRAL OPERATORS AS MULTIPLICATION OPERATORS AND PERTURBATION THEORY S. M. BAHRI 1 Abstract. I this paper we itroduce a multiplicatio
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationThe Poisson Summation Formula and an Application to Number Theory Jason Payne Math 248- Introduction Harmonic Analysis, February 18, 2010
The Poisso Summatio Formula ad a Applicatio to Number Theory Jaso Paye Math 48- Itroductio Harmoic Aalysis, February 8, This talk will closely follow []; however some material has bee adapted to a slightly
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More information