[...]... (a decreasing sequence, denoted {A,} 4 In the former case we define limn+m A, as u, ",, A,; in the latter, as n,"==,A, Then the countable additivity of measures also confers the following monotone property: (1) limnim p (A,) = p (A,) = p (limn+m A,) if {A,} JA,) if {A,} t and ( 2 ) 1-1 and p ( A l ) < x Thus, forfinite measures it is always true that limnioo p (A,) = p (limnim A,) A measure on (R 3)that... { J g , (x) dx};=: =, withg, (x)= ( l - l / n ) 1 [ 0 , 1 ] \ ~ (x) where l [ O l ~ \ ~takes the value unity on the irrationals of [ 0,1 ] Functions (x) {g,} are nonnegative and bounded by the integrable function 1[ 0,1 ~and (x ), g, (x) f g (x) = ~ [ o , ~ (x)for all x Thus, both theorems apply under the I\Q Lebesgueconstruction,andsolim ,, , J g , ( x ) d x = J 1 [ 0 , 1 1 \(x).dx = 1; ~ however, neither... z} ( a , b]-then lim Sn((a!bI)= n i m C g(zj)[F(Zj)-F(zj-)I ~.z~E(a,bl Note that when F has discontinuities at either a or b the limits of S ( ( a ,b ] ), , S ([a,) ) S ((a.) ) and S, ([a, will not all be the same Thus, the , b , b b]) notation Jab g dF is ambiguous when F has discontinuities, and to be explicit g.dF orJ(a,b)g.dF or Jla,bl g.dF as thespecific we write J(a,bl g d F or Jla,6) case... c R i.e ., but A E F,) Together these imply that R = A U A“ E F, that 0 = R“ E F, and that U:==,A, = Al U A2 U U A, E F for each finite n whenever all the {A3} are in F.Thus, fields are “closed” under complementation and finite unions = n;=lA; and (n;=lA3)c = U,”=,A,C-irnply that de Morgan’s laws-(Uy=,A,)“ closure extends to finite intersections as well For example, with R = 82 as our space, suppose... CALCULUS AND ANALYSIS 5 SRk represents k-dimensional Euclidean space 6 Symbol x represents Cartesian product Thus, SRk = 8 x 8 x x 8 7 Symbols ( a , b) [a,b ) , ( a ,b ] , [a b] indicate intervals of real number that are, respectively, open (not containing either endpoint ), left-closed, right-closed, and closed In such cases it is understood that a < b Thus, SR = (-co.oo) and SR+ = [ 0, m ) ,while (a, b)... limn too dx g (z).dxfor {a,} 4 integral b J, r Jay J!;n -ccoraslim ,, , [ 2 m g (x).dxfor{b,} t +moraslim ,, , g (z).dx However, these limits might well differ For example, if g (x) = x for all real x, then J a T g ( x ) dx = +cc for each n and so limn+m J f f T g ( x ) d x = +m, whereas limn+oo J2m (x) d x = limn-tm(-cc) g J g (x) dz = 0 The requirement that [ lgI ) ( : limn+= ,! Lebesgue integrability... increasing, right-continuous function F and its associated measure p ~ As in the construction of the Riemann integral, let a = z g < z < < 1 x,-1 < x, = b be a partition of [a, b] , and form the sum n where x; is an arbitrary point on [ x j - l ,x3]for each j If limnim and has the same value for each partition ( a b] for which lim max n+m.7€{1. 2, S ((a,] ) exists , b (z3-x3-1) = 0 ,n} and each... FROM CALCULUS AND ANALYSIS Alternatively, we could reach the same conclusion by applying the monotonicity property of A: X ( ( 0 , l ) )= X ( ( nlimo o O ~ ;] ) ,1 - - = hlX (( 0 , l - 3) hl(1 ;i) = - = 1 Thus, X ( ( O , l ] ) = X (( 0, l ) ) ,and so X ((1)) = X((O.l]\ ( 0 , l ) ) = X(( 0,1 ]) X ( ( 0 , l ) ) = 0 In words, the length measure associated with the singleton set (1) is zero Likewise, X ({x})... E 8, and so X ( [ a b ] ) = X ( [ a ,b ) ) = , X ( ( a ,b ] ) = X ( ( a ,b ) ) for all a b E 8 Moreover, X U (! ,: {xn}) = C = :X ({x~}) , = 0 for any countable number of points in 8 In particular, X(Q) = 0, X(!R\Q) = +m, and X([O l]\Q) = X ( [ 0 , 1 ] = 1 ) Let S E B be a measurable set in 8 such that X (S)= 0 If C is some condition or statement that holds for each real x except for points in S , then... Thus, if {A3}E1 E F then U,OO,,A, E F and nEIA, E F The field comprising finite unions of the intervals (a b] and (b m) in if2 is not a a field, since it does not contain finite open intervals (a,b ) It becomes a o field if we add in the countable unions, since (a,b) = Ur=? =, b - T I - ’ ] (a Of course, the collection of all subsets of a space is automatically a n field (if the space is finite, we . x0 y0 w0 h0" alt="" QUANTITATIVE FINANCE QUANTITATIVE FINANCE Its Development , Mat he mat i cal Foundations, and Current Scope T. W. Epps. Chapters 4, 5, 7, 1 0, 1 3, 1 8, and 23 are empirical projects that would be suitable for students with moderate computational skills and access to standard