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# Measure Theory and Advanced Probability ST 778

NCS

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This 8 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 778 at North Carolina State University taught by Montserrat Fuentes in Fall. Since its upload, it has received 16 views. For similar materials see /class/223945/st-778-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15

81 NS Fall 2007 Dr Fuentes Section l Preliminary Set Theory Let 2 be an abstrac set of elements eta In probability theory in general we consider the Set of all possible basic outcomes of an experiment and they will constitute a set Q known also as sample space Each basic ouzcome is denoted by to Let A C Q be a subset of It consists of all those elements it which belong to A Thus if the result of the expenmen leads to a basic outcome to which belongs to A Le if to E A then we say that the event A has occurred If the experimental result is an outcome g A then we say A Le complement of A has occurred Thus A9 w w e Q n 539 A Let B be asubset of A ie 8 C A Then any C 5 also belongs to A thus we say that the event 5 implies A Thereforeif B 2 then B C Let us now consider a family of sets or equivalently of events Let T be a fixed set called index set For example let T 1 2 n then we can construct a family of sets Consisting of A 1421 l l or we can denote this family by wnere we know the index set itquot The union of two Sets 4 and A2 written by A L Ag where E A or to E 8 including those w s which belong to both In the language of probability theory it may be described as the oceurrenoe of at least one of the events A or Ag Similarly the simultaneous occurrence of both events is denoted by the intersection of events A 51 A2 Note that A L Ag Ag L 11 A 2quot Ag 2 A f also A L 42 LJ A3 A1 Ll A2 L Ag These are known as commutative and associative laws and these laws also hold good for intersection ogeretlon also Two sets 41 an A2 are said to be disjoint if A 1 Ag does not have any point A set consisting of no points is coiled the nail set and is donotcd by 3 In probability theory disjoint sets are known mutually exclusive ovonts We also know A1 Iquot A1 A QAl 41 A A andif A C Ag then A L A z A and Al Ti A r A Further more it is easy to establish 41 Iquot L 41 A2 1quot Al 5 A3 anti A1UltA2 7 2 13 z A U A2 n A U AS A36 A1 is Q and S20 3 41 o Ag 2 o andif A c A2 then A6 3 Ag Important roiationships such as Demorgan s law can also be shown to be true A13 Aw A F A Ag L All these operations may be viewed as the laws of set operations with corresponding interpretation in the probabilistic language of events Even though we have defined unions and intersections of two sets it can be easily extended to any number of sets Thus if T be any arbitrary index sets than 1 Ag 1126A foraticastono 2567 EET a Aw wEAgforaiItET t T ThusifTL2n then on A1L3AQLJ itsquot 331d tQTAg 2 31375201330 An If T 123 than iii A t w 6 A for some f39migg n Nam that even though we wme U 121 we are not postulating the exxstence of n I a set A00 Slm arly 1An w w E An for all rms n m If T 0 11611 L A x I and A is defined to be Q tea I z 739 E T 0 A family 4263 is said to be a disjoint famiEy if Ail r2 Ag s and aim An r 432 is denoted by AglAgg in the case of disjaim famiiy U A will be denoted by Z A i E 3quot g g 3 Demorgzm s 13w holds goad for any index set T Le C C lt L a E Af and lt i Ag U 4 whether 3quot is t e T t e T t e T t e T finite countable infinity or uncountable infinity The difference betwesn two sets A and B denoted by A B maxi and wgBA ADBABCAQB B iv The symmetric difference two sets A and B is denoted by AAB A B o B A A8 o 346 Note AAA 3 A iB 2 821A and also A U SAC A U 8323A U 0 Limits and convergence for a sequence of sets A detour Let us first consider limits ofa sequence of real numbers Let on be a sequence of nite real numbers The sequence is said to be monotone increasing if unH 2 an Sometimes we distinguish between monotone and 39 3 A If nH 2 on then is monotone nondeoreasing denote it by 22 and if an gt up then on is monotone increasing denote it by on T Monotone increasing nondecreasing sequences always have a limit though it may be so In most nituations we denote o monotone increasing or monotone nondooreasing sequence on by the same notation on T A number is said to be least upper bound for a monotone oondocreasing sequence if for every e gt 0 there exists at loam one element of the Sequence say one no finite such that one 2 a e Note this statement implies that there is infinitely many 1th greater 1 6 since there has to be an element 2 o for every n A monotone nondecreasing sequence with a finite lub least upper bound has a limit and is equal to the least upper bound Similarly we can define a monotone decreasing nonincreasing sequence of real numbers and such asequence with a finite greatest lower bound hst a limit and is equal to the greatest lower bound New if we have an arbitrary sequence of real numbers m then from this arbitrary sequence we first construct the sequences based on the sequence an according to the following scheme ulzsu tzltumuqun wzihfu1mtzm ugsupuglze3t7m tuglhfuglumw 5 supttggtu4m 22R Supungiiz 1w 3 3 I mf m uYZI i quot Note that an is monotone honinereusing or decreasing whereas is moneiene nondecreasing or ineyeasing We shall indicate that by the notation an and 29 T Now if up has a finite greatest lower bound glb 253 then up i Similarly 7 240 where 1253 is the lub of sequence Now no 2 Kim u llm sup 22 and 183 lim iimiuf uni 1f ea 2 we then the common value is called the limit of the arbitrary sequence ht fact an arbitrary sequence of real numbers can have many limit pointsv All the limit points will lie between iim supun and lim inf u So if these two limits are equal then all the possible limits points will have the same value and in that case the common value is called the Sim up Note that lim sup tin and lim infun always exist though they may not be equal When they are not equal then the limit of the sequence does not exist Limit of a Sgguencc of sets Consider a sequence of subsets An This sequence is said to be monotone mendecreas tng if A lt A2 A3 g The limit ofthis sequence of sets is given by D An Note that the knitting set U An 15 a subset of 2 n 1 n 1 A point we kl A if we some Am where m is a finite integer But since An T then 231524 for all 36 7mm 1 Thatis each as in theiimiting set belongs to aii but possibly a finite number of An and hence w belongs to infinitely many sets ofttxe sequence An Similarly the sequence An is said to be monotone nonincreasing if A 2 Ag 2 A3 2 A 2 denoted by i A 00 An L t he limit ofthe sequence of sets IS given by H 14 In this case a w 71 belonging to oniy a finite number of sets in the sequence cannot belong to the limiting set So for monotone sequences the limiting set is well defined Consider now an arbitrary sequence of sets Ag Again we shah construct from this arbitrary sequence two other sequences of subsets of Q which wilt be one 39 and the other monotone nonincreasing Le 23 CL A Ci 0 AI 7 i n 1 B n 2A C2 Ln 2An 13 fin Q Ci An Note that BR is monotone nonincreasing as Bi 2 82 Q 83 2 and similariy CE is menoione nandecmasing QC 00 The 1mm of Ba u 13 incl mt Am andthisisknownas n n quot 06 limsup An Simiiarlythelimitof an 33 CH 3 Am and is I x 32 nxlm known as lim inf An Now if these two limiting sets are equal than this common setis known as inn An Notethatif AHT than 321 513 C l an also an uCn r Mn n I So for A limsup an Eimiann iii Similarly if Am i then Bn A lim Bn 2 9an 3 14 n x 7 OC QC 00 0 13 and CR 1 Ag 3 Ag UCn 1397 An n z 1 n 1 n 4 oo hm sup An 1m mf ln f A n 1 Note that every point 25 5 f2 may be classi ed according to the faZEowing scheme 1 w maynotbelongto any oftne subsets An 392 maybclongzo only nitely many subsets An 3 u may balong to infinitely many subsets ofthc sequence of An and may m3 beieng to infinitely many subsets of the sequence AH As an exampie maybelongto A143A3 Agn and 2 may not beiong to 12144 915 Azm 421 may belong to in nitely many subsets of An and may mt beiong to only niter many subsets of An New lim sup 4 consisLs of an those points which belongs t0 infinitely many An it consists of those which beiong to category 3 and 4 where as 11m inf An consists of 31 those w s which belcmg to ail but a nite number of the sequence of sets An and so belong 0 category 4 Now if w e liminf n than 10 Em for some nite no a u must belong to infiniter many An s Hence w 6 lim sup An Hence Iim inf13 C lim sup AR Examp e Consider two subsets B and 3 Let Agn1z8 for n012 znzcquot V cc 7 AmzsuC iimsupAmm n a NAmBUC Wl n 33 Now Am8rwc liminf Cri AmZBmC m n nzlmmn 39 in this case lim sup An lint inf An unless B C Now if 8 H C x if then iim sup 4 B u C and lim inf An

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