PROBABILITY FOR STATISTICIANS

By MATTHEW GONZALEZ

CONTENTS CHAPTER 1 MEASURES 1 BASIC PROPERTIES OF MEASURES 2 CONSTRUCTION AND EXTENSION OF MEASURES 3 LEBESGUE–STIELTJES MEASURES CHAPTER 2 MEASURABLE FUNCTIONS AND CONVERGENCE 1 MAPPINGS AND ?-FIELDS 2 MEASURABLE FUNCTIONS 3 CONVERGENCE 4 PROBABILITY, RVS, AND CONVERGENCE IN LAW 5 DISCUSSION OF SUB ?-FIELDS CHAPTER 3 INTEGRATION 1 THE LEBESGUE INTEGRAL 2 FUNDAMENTAL PROPERTIES OF INTEGRALS 3 EVALUATING AND DI?ERENTIATING INTEGRALS 4 INEQUALITIES 5 MODES OF CONVERGENCE CHAPTER 4 DERIVATIVES VIA SIGNED MEASURES 1 DECOMPOSITION OF SIGNED MEASURES 2 THE RADON–NIKODYM THEOREM 3 LEBESGUE’S THEOREM 4 THE FUNDAMENTAL THEOREM OF CALCULUS CHAPTER 5 MEASURES AND PROCESSES ON PRODUCTS 1 FINITE-DIMENSIONAL PRODUCT SPACES 2 RANDOM VECTORS ON (?, A, P) 3 COUNTABLY IN?NITE PRODUCT PROBABILITY SPACES 4 RANDOM ELEMENTS AND PROCESSES ON (?, A, P) CHAPTER 6 GENERAL TOPOLOGY AND HILBERT SPACE 1 GENERAL TOPOLOGY 2 METRIC SPACES 3 HILBERT SPACE CHAPTER 7 DISTRIBUTION AND QUANTILE FUNCTIONS 1 CHARACTER OF DISTRIBUTION FUNCTIONS 2 PROPERTIES OF DISTRIBUTION FUNCTIONS 3 THE QUANTILE TRANSFORMATION 4 INTEGRATION BY PARTS APPLIED TO MOMENTS 5 IMPORTANT STATISTICAL QUANTITIES 6 IN?NITE VARIANCE 7 SLOWLY VARYING PARTIAL VARIANCE 8 SPECI?C TAIL RELATIONSHIPS 9 REGULARLY VARYING FUNCTIONS 10 SOME WINSORIZED VARIANCE COMPARISONS 11 INEQUALITIES FOR WINSORIZED QUANTILE FUNCTIONS CHAPTER 8 INDEPENDENCE AND CONDITIONAL DISTRIBUTIONS 1 INDEPENDENCE 2 THE TAIL ?-FIELD 3 UNCORRELATED RANDOM VARIABLES 4 BASIC PROPERTIES OF CONDITIONAL EXPECTATION 5 REGULAR CONDITIONAL PROBABILITY 6 CONDITIONAL EXPECTATIONS AS PROJECTIONS CHAPTER 9 SPECIAL DISTRIBUTIONS 1 ELEMENTARY PROBABILITY 2 DISTRIBUTION THEORY FOR STATISTICS 3 LINEAR ALGEBRA APPLICATIONS 4 THE MULTIVARIATE NORMAL DISTRIBUTION CHAPTER 10 WLLN, SLLN, LIL, AND SERIES 0 INTRODUCTION 1 BOREL–CANTELLI AND KRONECKER LEMMAS 2 TRUNCATION, WLLN, AND REVIEW OF INEQUALITIES 3 MAXIMAL INEQUALITIES AND SYMMETRIZATION 4 THE CLASSICAL LAWS OF LARGE NUMBERS, LLNS 5 APPLICATIONS OF THE LAWS OF LARGE NUMBERS 6 GENERAL MOMENT ESTIMATION 7 LAW OF THE ITERATED LOGARITHM 8 STRONG MARKOV PROPERTY FOR SUMS OF IID RVS 9 CONVERGENCE OF SERIES OF INDEPENDENT RVS 10 MARTINGALES 11 MAXIMAL INEQUALITIES, SOME WITH ?BOUNDARIES 12 A UNIFORM SLLN CHAPTER 11 CONVERGENCE IN DISTRIBUTION 1 STEIN’S METHOD FOR CLTS 2 ˜WINSORIZATION AND ?TRUNCATION 3 IDENTICALLY DISTRIBUTED RVS 4 BOOTSTRAPPING 5 BOOTSTRAPPING WITH SLOWLY INCREASING TRIMMING 6 EXAMPLES OF LIMITING DISTRIBUTIONS 7 CLASSICAL CONVERGENCE IN DISTRIBUTION 8 LIMIT DETERMINING CLASSES OF FUNCTIONS CHAPTER 12 BROWNIAN MOTION AND EMPIRICAL PROCESSES 1 SPECIAL SPACES 2 EXISTENCE OF PROCESSES ON (C, C) AND (D, D) 3 BROWNIAN MOTION AND BROWNIAN BRIDGE 4 STOPPING TIMES 5 STRONG MARKOV PROPERTY 6 EMBEDDING A RV IN BROWNIAN MOTION 7 BARRIER CROSSING PROBABILITIES 8 EMBEDDING THE PARTIAL SUM PROCESS 9 OTHER PROPERTIES OF BROWNIAN MOTION 10 VARIOUS EMPIRICAL PROCESSES 11 INEQUALITIES FOR THE VARIOUS EMPIRICAL PROCESSES 12 APPLICATIONS CHAPTER 13 CHARACTERISTIC FUNCTIONS 1 BASIC RESULTS, WITH DERIVATION OF COMMON CHFS 2 UNIQUENESS AND INVERSION 3 THE CONTINUITY THEOREM 4 ELEMENTARY COMPLEX AND FOURIER ANALYSIS 5 ESSEEN’S LEMMA 6 DISTRIBUTIONS ON GRIDS 7 CONDITIONS FOR ? TO BE A CHARACTERISTIC FUNCTION CHAPTER 14 CLTS VIA CHARACTERISTIC FUNCTIONS 0 INTRODUCTION 1 BASIC LIMIT THEOREMS 2 VARIATIONS ON THE CLASSICAL CLT 3 LOCAL LIMIT THEOREMS 4 GAMMA APPROXIMATION 5 EDGEWORTH EXPANSIONS 6 APPROXIMATING THE DISTRIBUTION OF H( ¯XN) CHAPTER 15 IN?NITELY DIVISIBLE AND STABLE DISTRIBUTIONS 1 IN?NITELY DIVISIBLE DISTRIBUTIONS 2 STABLE DISTRIBUTIONS 3 CHARACTERIZING STABLE LAWS 4 THE DOMAIN OF ATTRACTION OF A STABLE LAW CHAPTER 16 ASYMPTOTICS VIA EMPIRICAL PROCESES 0 INTRODUCTION 1 TRIMMED AND WINSORIZED MEANS 2 LINEAR RANK STATISTICS AND FINITE SAMPLING 3 THE BOOTSTRAP 4 L-STATISTICS CHAPTER 17 ASYMPTOTICS VIA STEIN’S APPROACH 1 U-STATISTICS 2 HOE?DING’S COMBINATORIAL CLT CHAPTER 18 MARTINGALES 1 BASIC TECHNICALITIES FOR MARTINGALES 2 SIMPLE OPTIONAL SAMPLING THEOREM 3 THE SUBMARTINGALE CONVERGENCE THEOREM 4 APPLICATIONS OF THE S-MG CONVERGENCE THEOREM 5 DECOMPOSITION OF A SUBMARTINGALE SEQUENCE 6 OPTIONAL SAMPLING 7 APPLICATIONS OF OPTIONAL SAMPLING 8 INTRODUCTION TO COUNTING PROCESS MARTINGALES 9 DOOB–MEYER SUBMARTINGALE DECOMPOSITION 10 PREDICTABLE PROCESSES AND 0 H DM MARTINGALES 11 THE BASIC CENSORED DATA MARTINGALE 12 CLTS FOR DEPENDENT RVS CHAPTER 19 CONVERGENCE IN LAW ON METRIC SPACES 1 CONVERGENCE IN DISTRIBUTION ON METRIC SPACES 2 METRICS FOR CONVERGENCE IN DISTRIBUTION CHAPTER MEASURES 2M ?F(X) < K 2M (3) CHAPTER 1 MEASURES CAN BE SPECI?ED THIS APPROACH LEADS TO THE CONCEPT OF A ?-?ELD A OF SUBSETS OF [A, B] THAT ARE MEASURABLE (THAT IS, WE MUST BE ABLE TO ASSIGN TO THESE INTERVALS A NUMBER CALLED THEIR “LENGTH”), AND THIS LEADS TO THE CONCEPT OF THE CLASS M OF MEASURABLE FUNCTIONS THIS CLASS M OF MEASURABLE FUNCTIONS WILL BE SEEN TO BE CLOSED UNDER PASSAGE TO THE LIMIT AND ALL THE OTHER OPERATIONS THAT WE ARE ACCUSTOMED TO PER- FORMING ON FUNCTIONS MOREOVER, THE DESIRABLE PROPERTY B A FN(X) DX ? B A F(X) DX FOR FUNCTIONS FN “CONVERGING” TO F WILL BE BROADLY TRUE XMI XMM … … THE DOMAIN OF F( ) IS EQUALLY DIVIDED RIEMANN SUMS THE RANGE OF F( ) IS EQUALLY DIVIDED LEBESGUE SUMS FIGURE 1 1 RIEMANN SUMS AND LEBESGUE SUMS 2M 1 BASIC PROPERTIES OF MEASURES DE?NITION 1 1 (SET THEORY) CONSIDER A NONVOID CLASS A OF SUBSETS A OF A NONVOID SET ? (FOR US, ?WILL BE THE SAMPLE SPACE OF AN EXPERIMENT ) (A) LET AC DENOTE THE COMPLEMENT OF A, LET A ?B DENOTE THE UNION OF A AND B, LET A ?B AND AB BOTH DENOTE THE INTERSECTION, LET A \ B ?ABC DENOTE THE SET DI?ERENCE, LET A?B ?(ACB ?ABC) DENOTE THE SYMMETRIC DI?ERENCE, AND LET ? DENOTE THE EMPTY SET THE CLASS OF ALL SUBSETS OF ?WILL BE DENOTED BY 2? SETS A AND B ARE CALLED DISJOINT IF AB = ?, AND SEQUENCES OF SETS AN OR CLASSES OF SETS AT ARE CALLED DISJOINT IF ALL PAIRS ARE DISJOINT WRITING A+B OR ? 1 AN WILL ALSO DENOTE A UNION, BUT WILL IMPLY THE DISJOINTNESS OF THE SETS IN THE UNION AS USUAL, A ?B DENOTES THAT A IS A SUBSET OF B WE CALL A SEQUENCE AN INCREASING (AND WE WILL NEARLY ALWAYS DENOTE THIS FACT BY WRITING AN ?) WHEN AN ?AN+1 FOR ALL N ?1 WE CALL THE SEQUENCE DECREASING (DENOTED BY AN ?) WHEN AN ?AN+1 FOR ALL N ?1 WE CALL THE SEQUENCE MONOTONE IF IT IS EITHER INCREASING OR DECREASING LET ? DENOTE A GENERIC ELEMENT OF ? WE WILL USE 1A( ) TO DENOTE THE INDICATOR FUNCTION OF A, WHICH EQUALS 1 OR 0 AT ? ACCORDING AS ? ?A OR ? ??A (B) A WILL BE CALLED A ?ELD IF IT IS CLOSED UNDER COMPLEMENTS AND UNIONS (THAT IS, A AND B IN A REQUIRES THAT AC AND A ?B BE IN A ) [NOTE THAT BOTH ?AND ?ARE NECESSARILY IN A, AS A WAS ASSUMED TO BE NONVOID, WITH ?= A ?AC AND ?= ?C ] (C) A WILL BE CALLED A ?-?ELD IF IT IS CLOSED UNDER COMPLEMENTS AND COUNTABLE UNIONS (THAT IS, A, A1, A2, IN A REQUIRES THAT AC AND ?? 1 AN BE IN A ) (D) A WILL BE CALLED A MONOTONE CLASS PROVIDED IT CONTAINS ?? 1 AN FOR ALL INCREASING SEQUENCES AN IN A AND CONTAINS ?? 1 AN FOR ALL DECREASING SEQUENCES AN IN A (E) (?, A) WILL BE CALLED A MEASURABLE SPACE PROVIDED A IS A ?-?ELD OF SUBSETS OF ? (F) A WILL BE CALLED A ?-SYSTEM PROVIDED AB IS IN A FOR ALL A AND B IN A; AND A WILL BE CALLED A ¯?-SYSTEM WHEN ?IN A IS ALSO GUARANTEED IF A IS A ?ELD (OR A ?-?ELD), THEN IT IS CLOSED UNDER INTERSECTIONS (UNDER COUNTABLE INTERSECTIONS); SINCE AB = (AC ?BC)C (SINCE ?? 1 AN = (?? 1 AC N)C) LIKEWISE, WE COULD HAVE USED “INTERSECTION” INSTEAD OF “UNION” IN OUR DE?NITIONS BY MAKING USE OF A ?B = (AC ?BC)C AND ?? 1 AN = (?? 1 AC N)C PROPOSITION 1 1 (CLOSURE UNDER INTERSECTIONS) (A) ARBITRARY INTERSECTIONS OF ?ELDS, ?-?ELDS, OR MONOTONE CLASSES ARE ?ELDS, ?-?ELDS, OR MONOTONE CLASSES, RESPECTIVELY [FOR EXAMPLE, ?{F? : F? IS A ?ELD UNDER CONSIDERATION} IS A ?ELD ] (B) THERE EXISTS A MINIMAL ?ELD, ?-?ELD, OR MONOTONE CLASS GENERATED BY (OR, CON- TAINING) ANY SPECI?ED CLASS C OF SUBSETS OF ? WE CALL C THE GENERATORS FOR EXAMPLE, ?[C] ?{F? : F? IS A ?-?ELD OF SUBSETS OF ?FOR WHICH C ?F?} (4) IS THE MINIMAL ?-?ELD GENERATED BY C (THAT IS, CONTAINING C) (C) A COLLECTION A OF SUBSETS OF ?IS A ?-?ELD IF AND ONLY IF IT IS BOTH A ?ELD AND A MONOTONE CLASS PROOF (C) (?) ?? 1 AN = ?? 1 (?N 1 AK)) ??? 1 BN ?A SINCE THE BN ARE IN A AND ARE ? EVERYTHING ELSE IS EVEN MORE TRIVIAL µ(AN) FOR ALL DISJOINT SEQUENCES AN IN A, (5) THEN µ IS CALLED A MEASURE (OR, EQUIVALENTLY, A COUNTABLY ADDITIVE MEASURE) ON (?, A) THE TRIPLE (?, A, µ) IS THEN CALLED A MEASURE SPACE WE CALL µ ?NITE IF µ(?) < ? WE CALL µ ?-?NITE IF THERE EXISTS A MEASURABLE DECOMPOSITION OF ?AS ?= ? 1 ?N WITH ?N ?A AND µ(?N) < ?FOR ALL N [IF A IS NOT A ?-?ELD, WE WILL STILL CALL µ A MEASURE ON (?, A), PROVIDED THAT (5) HOLDS FOR ALL SEQUENCES AN FOR WHICH ?? 1 AN IS IN A WE WILL NOT, HOWEVER, USE THE TERM “MEASURE SPACE” TO DESCRIBE SUCH A TRIPLE WE WILL CONSIDER BELOW MEASURES ON ?ELDS, ON CERTAIN ¯?-SYSTEMS, AND ON SOME OTHER COLLECTIONS OF SETS A USEFUL PROPERTY OF A COLLECTION OF SETS IS THAT ALONG WITH ANY SETS A1, , AK IT ALSO INCLUDE ALL SETS OF THE TYPE BK ?AKAC K?1 AC 2AC 1; THEN N 1AN = N 1BK IS ALSO INCLUDED ] (B) WE CALL µ A ?NITELY ADDITIVE MEASURE (ABBREVIATED F A ) ON (?, A) IF µ( N 1AN) = N 1µ(AN) (6) FOR ALL DISJOINT SEQUENCES AN IN A FOR WHICH N 1AN IS ALSO IN A DE?NITION 1 3 (OUTER MEASURES) CONSIDER A SET FUNCTION µ?: 2??[0, ?] (A) SUPPOSE THAT µ?SATIS?ES THE FOLLOWING THREE PROPERTIES NULL: µ?(?) = 0 MONOTONE: µ?(A) ?µ?(B) FOR ALL A ?B COUNTABLE SUBADDITIVITY: µ?(? 1 AN) ?? 1 µ?(AN) FOR ALL AN THEN µ?IS CALLED AN OUTER MEASURE (B) AN ARBITRARY SUBSET A OF ?IS CALLED µ?-MEASURABLE IF µ?(T) = µ?(TA) + µ(TAC) FOR ALL SUBSETS T ?? (7) SETS T USED IN THIS CAPACITY ARE CALLED TEST SETS (C) WE LET A?DENOTE THE CLASS OF ALL µ?-MEASURABLE SETS, THAT IS, A??{A ?2?: A IS µ?-MEASURABLE} (8) [NOTE THAT A ?A?IF AND ONLY IF µ?(T) ?µ?(TA) + µ?(TAC) FOR ALL T ??, SINCE THE OTHER INEQUALITY IS TRIVIAL BY THE SUBADDITIVITY OF µ? ] MOTIVATION 1 2 (MEASURE) IN THIS PARAGRAPH WE WILL CONSIDER ONLY ONE POSSIBLE MEASURE µ, NAMELY THE LEBESGUE-MEASURE GENERALIZATION OF LENGTH LET CI DENOTE THE SET OF ALL INTERVALS OF THE TYPES (A, B], (??, B], AND (A, +?) ON THE REAL LINE R, AND FOR EACH OF THESE INTERVALS I WE ASSIGN A MEASURE VALUE µ(I) EQUAL TO ITS LENGTH, THUS ?, B ?A, ?IN THE THREE SPECIAL CASES ALL IS WELL UNTIL WE MANIPULATE THE SETS 1 BASIC PROPERTIES OF MEASURES IN CI, AS EVEN THE UNION OF TWO ELEMENTS IN CI NEED NOT BE IN CI THUS, CI IS NOT A VERY RICH COLLECTION OF SETS A NATURAL EXTENSION IS TO LET CF DENOTE THE COLLECTION OF ALL ?NITE DISJOINT UNIONS OF SETS IN CI, WHERE THE MEASURE µ(A) WE ASSIGN TO EACH SUCH SET A IS JUST THE SUM OF THE MEASURES (LENGTHS) OF ALL ITS DISJOINT PIECES NOW CF IS A ?ELD, AND IS THUS CLOSED UNDER THE ELEMENTARY OPERATIONS OF UNION, INTERSECTION, AND COMPLEMENTATION MUCH CAN BE DONE USING ONLY CF AND LETTING “MEASURE” BE THE “EXACT LENGTH” BUT CF IS NOT CLOSED UNDER PASSAGE TO THE LIMIT, AND IT IS THUS INSU?CIENT FOR MANY OF OUR NEEDS FOR THIS REASON THE CONCEPT OF THE SMALLEST ?-?ELD CONTAINING CF , LABELED B ??[CF ], IS INTRODUCED WE CALL B THE BOREL SETS BUT LET US WORK BACKWARDS LET US ASSIGN AN OUTER MEASURE VALUE µ?(A) TO EVERY SUBSET A IN THE CLASS 2R OF ALL SUBSETS OF THE REAL LINE R IN PARTICULAR, TO ANY SUBSET A WE ASSIGN THE VALUE µ?(A) THAT IS THE IN?MUM OF ALL POSSIBLE NUMBERS ? N=1 µ(AN), IN WHICH EACH AN IS IN THE ?ELD CF (SO THAT WE KNOW ITS MEASURE) AND IN WHICH THE AN’S FORM A COVER OF A (IN THAT A ??? 1 AN) THUS EACH NUMBER ? 1 µ(AN) IS A NATURAL UPPER BOUND TO THE MEASURE (OR GENERALIZED LENGTH) OF THE SET A, AND WE WILL SPECIFY THE IN?MUM OF SUCH UPPER BOUNDS TO BE THE OUTER MEASURE OF A THUS TO EACH SUBSET A OF THE REAL LINE WE ASSIGN A VALUE µ?(A) OF GENERALIZED LENGTH THIS VALUE SEEMS “REASONABLE”, BUT DOES IT “PERFORM CORRECTLY”? LET US SAY THAT A PARTICULAR SET A IS µ?-MEASURABLE (THAT IS, IT “PERFORMS CORRECTLY”) IF µ?(T) = µ?(TA) + µ?(TAC) FOR ALL SUBSETS OF THE REAL LINE R, THAT IS, IF THE A VERSUS AC DIVISION OF THE LINE DIVIDES EVERY SUBSET T OF THE LINE INTO TWO PIECES IN A FASHION THAT IS µ?-ADDITIVE THIS IS UNDOUBTEDLY A COMBINATION OF REASONABLENESS AND ?NE TECHNICALITY THAT TOOK SOME TIME TO EVOLVE IN THE MIND OF ITS CREATOR, CARATH´EODORY, WHILE HE SEARCHED FOR A CONDITION THAT “WORKED” IN WHAT SENSE DOES IT “WORK”? THE COLLECTION A?OF ALL µ?-MEASURABLE SETS TURNS OUT TO BE A ?-?ELD THUS THE COLLECTION A?IS CLOSED UNDER ALL OPERATIONS THAT WE ARE LIKELY TO PERFORM; AND IT IS BIG ENOUGH, IN THAT IT IS A ?-?ELD THAT CONTAINS CF THUS WE WILL WORK WITH THE RESTRICTION µ?|A?OF µ?TO THE SETS OF A?(HERE, THE VERTICAL LINE MEANS “RESTRICTED TO”) THIS IS ENOUGH TO MEET OUR NEEDS THERE ARE MANY MEASURES OTHER THAN LENGTH FOR AN ?AND RIGHT-CONTINUOUS FUNCTION F ON THE REAL LINE (CALLED A GENERALIZED DF) WE DE?NE THE STIELTJES MEASURE OF AN ARBITRARY INTERVAL (A, B] (WITH ???A < B ??) IN CI BY µF ((A, B]) = F(B)?F(A), AND WE EXTEND IT TO SETS IN CF BY ADDING UP THE MEASURE OF THE PIECES REAPPLYING THE PREVIOUS PARAGRAPH, WE CAN EXTEND µF TO THE µ? F -MEASURABLE SETS IT IS THE IMPORTANT CARATH´EODORY EXTENSION THEOREM THAT WILL ESTABLISH THAT ALL STIELTJES MEASURES (INCLUDING THE CASE OF ORDINARY LENGTH, WHERE F(X) = X AS CONSIDERED IN THE ?RST PARAGRAPH) CAN BE EXTENDED FROM CF TO THE BOREL SETS B THAT IS, ALL BOREL SETS ARE µ?-MEASURABLE FOR EVERY STIELTJES MEASURE ONE FURTHER EXTENSION IS POSSIBLE, IN THAT EVERY MEASURE CAN BE “COMPLETED” (SEE THE END OF SECTION 1 2) WE NOTE HERE ONLY THAT WHEN THE STIELTJES MEASURE µF ASSOCIATED WITH THE GENERALIZED DF F IS “COMPLETED”, ITS DOMAIN OF DE?NITION IS EXTENDED FROM THE BOREL SETS B (WHICH ALL STIELTJES MEASURES HAVE IN COMMON) TO A LARGER COLLECTION ˆBµF THAT DEPENDS ON THE PARTICULAR F IT IS LEFT TO SECTION 1 2 TO SIMPLY STATE THAT THIS IS AS FAR AS WE CAN GO THAT IS, EXCEPT IN TRIVIAL SPECIAL CASES, WE ?ND THAT ˆBµF IS A PROPER SUBSET OF 2R (THAT IS, IT IS TYPICALLY IMPOSSIBLE TO TRY TO DE?NE THE MEASURE OF ALL SUBSETS OF ?IN A SUITABLE FASHION ) CHAPTER 1 MEASURES EXAMPLE 1 1 (SOME EXAMPLES OF MEASURES, INFORMALLY) (A) LEBESGUE MEASURE: LET ?(A) DENOTE THE LENGTH OF A (B) COUNTING MEASURE: LET #(A) DENOTE THE NUMBER OF “POINTS” IN A (OR THE CARDINALITY OF A) (C) UNIT POINT MASS: LET ??0(A) ?1{?0}(A), ASSIGNING MEASURE 1 OR 0 AS ?0 ?A OR NOT EXAMPLE 1 2 (BOREL SETS) (A) LET ?= R AND LET C CONSIST OF ALL ?NITE DISJOINT UNIONS OF INTERVALS OF THE TYPES (A, B], (??, B], AND (A, +?) CLEARLY, C IS A ?ELD THEN B ??[C] WILL BE CALLED THE BOREL SETS (OR THE BOREL SUBSETS OF R) LET µ(A) BE DE?NED TO BE THE SUM OF THE LENGTHS OF THE INTERVALS COMPOSING A, FOR EACH A ?C THEN µ IS A C A MEASURE ON THE ?ELD C, AS WILL BE SEEN IN THE PROOF OF THEOREM 1 3 1 BELOW (B) IF (?, D) IS A METRIC SPACE AND U ?{ALL D-OPEN SUBSETS OF ?}, THEN B ??[U] WILL BE CALLED THE BOREL SETS OR THE BOREL ?-?ELD (C) IF (?, D) IS (R, | |) FOR ABSOLUTE VALUE | |, THEN ?[C] = ?[U] EVEN THOUGH C ?= U [THIS CLAIM IS TRUE, SINCE C ??[U] AND U ??[C] ARE CLEAR THEN, JUST MAKE A TRIVIAL APPEAL TO EXERCISE 1 1 ] (D) LET ¯R ?[??, = ?] DENOTE THE EXTENDED REAL LINE AND LET ¯B ??[B, {??}, {+?}] PROPOSITION 1 2 (MONOTONE PROPERTY OF MEASURES) LET (?, A, µ) DENOTE A MEASURE SPACE LET (A1, A2, ) BE IN A (A) IF AN ?AN+1 FOR ALL N, THEN µ( ? 1 AN) = LIM N??µ(AN) (9) (B) IF µ(AN0) < ?FOR SOME N0, AND AN ?AN+1 FOR ALL N, THEN µ( ? N=1AN) = LIM N??µ(AN) (10) [LETTING ?DENOTE THE REAL LINE R, LETTING AN = [N, ?), AND LETTING µ DENOTE EITHER LEBESGUE MEASURE OR COUNTING MEASURE, WE SEE THE NEED FOR SOME REQUIREMENT ] (C) (COUNTABLE SUBADDITIVITY) WHENEVER (A1, A2, ) AND ?? 1 AN ARE ALL IN A, THEN µ( ? 1 AK) ?? 1 µ(AK) ; AND THIS ALSO HOLDS TRUE FOR A MEASURE ON A ?ELD OR ON A ¯?-SYSTEM PROOF (A) NOW, µ( ?? 1 AN) = µ( ? 1 (AN \ AN?1)) WITH A0 ?? = ? 1 µ(AN \ AN?1) BY C A (P) = LIM N N K=1 µ(AK \ AK?1) = LIM N µ( N K=1 (AK \ AK?1)) BY F A = LIM N µ(AN) (Q) 1 BASIC PROPERTIES OF MEASURES (B) WITHOUT LOSS OF GENERALITY, REDE?NE A1 = A2 = = AN0 LET BN ?A1\AN, SO THAT BN ? THUS, ON THE ONE HAND, LIM N µ(BN) = µ( ?? 1 BN) BY (A) = µ( ?? 1 (A1 ?AC N)) = µ(A1 ??? 1 AC N) = µ(A1 ?(?? 1 AN)C) = µ(A1) ?µ( ?? 1 AN) (R) ON THE OTHER HAND, LIM N µ(BN) = LIM N { µ(A1) ?µ(AN)} BY F A = µ(A1) ?LIM N µ(AN) (S) EQUATE (R) AND (S); SINCE µ(A1) < ?, WE CAN CANCEL IT TO OBTAIN THE EQUALITY µ(?? 1 AN) = LIMN µ(AN) (C) LET B1 ?A1, B2 ?A2AC 1, , BK ?AKAC K?1 AC 1 THEN THESE NEWLY DE?NED SETS BK ARE DISJOINT, AND ?N K=1AK = N K=1 BK HENCE [A TECHNIQUE WORTH REMEMBERING] µ( N K=1 AK) = µ( N K=1 BK) = N K=1 µ(BK) WHERE N K=1AK = N K=1 BK FOR BK ?AKAC K?1 AC (11) ?N 1µ(AK) ?? 1 µ(AK) BY MONOTONICITY (T) LET N ??, AND USE PART (A) TO GET THE RESULT DE?NITION 1 4 (CONTINUITY OF MEASURES) A MEASURE µ IS CONTINUOUS FROM BELOW (ABOVE) IF µ(LIM AN) = LIM µ(AN) FOR ALL AN ?(FOR ALL AN ?, WITH AT LEAST ONE µ(AN) ?NITE) WE CALL µ CONTINUOUS IN CASE IT IS CONTINUOUS BOTH FROM BELOW AND FROM ABOVE THE NEXT RESULT IS OFTEN USED IN CONJUNCTION WITH THE CARATH´EODORY EXTENSION THEOREM OF THE NEXT SECTION VIEW IT AS A CONVERSE TO THE PREVIOUS PROPOSITION PROPOSITION 1 3 (CONTINUITY OF MEASURES) IF A ?NITELY ADDITIVE MEASURE µ ON EITHER A ?ELD OR ?-?ELD IS EITHER CONTINUOUS FROM BELOW OR HAS µ(?) < ?AND IS CONTINUOUS FROM ABOVE AT ?, THEN IT IS A COUNTABLY ADDITIVE MEASURE CHAPTER 1 MEASURES PROOF SUPPOSE ?RST THAT µ IS CONTINUOUS FROM BELOW THEN µ( ? 1 AN) = µ( LIMN 1AK) = LIM µ( N 1AK) BY CONTINUITY FROM BELOW (A) = LIM N 1 µ(AK) BY F A (WHERE WE USED ONLY THAT A IS A ?ELD) (B) = ? 1 µ(AN), (C) GIVING THE REQUIRED COUNTABLE ADDITIVITY SUPPOSE NEXT THAT µ IS ?NITE AND IS ALSO CONTINUOUS FROM ABOVE AT ? THEN F A (EVEN IF A IS ONLY A ?ELD) GIVES µ( ? 1 AN) = µ( N 1AK) + µ( ? N+1AK) = N 1 µ(AK) + µ( ? N+1AK) ?? 1 µ(AK) + 0, (D) WHERE µ(? N+1 AK) ?µ(?) = 0 BY CONTINUITY FROM ABOVE AT ?, SINCE ? N+1 AK ?? AND µ IS ?NITE SUPK?N AK , (15) AND THESE YIELD THE SMALLEST LIMIT POINT AND THE LARGEST LIMIT POINT, RESPECTIVELY, OF THE SEQUENCE AN 1 BASIC PROPERTIES OF MEASURES