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Financial Mathematics 378 > Sets, logic, measures & measurable functions > Flashcards

Sets, logic, measures & measurable functions Flashcards

1
Q

Union: A or B

A

{x: x element of A or x element of B}

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2
Q

Intersection: A and B

A

={x: x element of A and x element of B}

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3
Q

Infinite union: U(Ai element of S}

A

={x: exists an i element of I (x element of Ai)}

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4
Q

Infinite intersection: N(Ai element of S)

A

={x: for all i element of I (x element of Ai)}

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5
Q

Set difference: A-B

A

=A - (A n B) = A n B^c

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6
Q

Symmetric Difference: A triangle B

A

=(A or B) - (A n B)
=(A-B) or (B-A)
=(A n B^c) or (A^c n B)

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7
Q

(A triangle B)^c

A

= A^c n B

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8
Q

Triangular inequality

A

=(A triangle C) subset (A triangle B) or (B triangle C)

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9
Q

A triangle C =

A

=(A triangle B) triangle (B triangle C)

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10
Q

When is a set A countable?

A

= if set A is finite or there is a 1-to-1 correspondence with the set of natural numbers
= there is a bijection from A to a subset of natural numbers

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11
Q

Push-forward function f: X -> Y, what is f(A)?

A

f(A) = {f(x) | x element of A}
~DONT preserve set operations

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12
Q

Push-back function f^-1: Y -> X, what is f^-1(A)?

A

f^-1(A) = {x element of A| f(x) element of B}
~DOES preserve set operations

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13
Q

max(A) = M <=>

A

for all x’s element of A (x ≤ M) and M element of A

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14
Q

min(A) = m <=>

A

for all x’s element of A (X ≥ m) and m element of A

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15
Q

Does min(A) and max(A) always exist?

A

No

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16
Q

sup(A) = M <=>

A

for all x’s element of A (x ≤ M) and M is the smallest of real numbers that satisfy this condition

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17
Q

inf(A) = m <=>

A

for all x’s element of A (x ≥ m) and m is the largest of real numbers that satisfy this condition

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18
Q

Does sup(A) and inf(A) always exist?

A

Yes

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19
Q

In laymen’s terms what is sup(A)?

A

its least UB

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20
Q

In laymen’s terms what is inf(A)?

A

its greatest LB

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21
Q

sup(empty set) =

A

infinity

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22
Q

inf(empty set) =

A

-infinity

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23
Q

If A is unbounded: sup(A) =

A

infinity

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24
Q

If A is unbounded: inf(A) =

A

-infinity

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25
Q

limnsup(An) =

A

={x element of OMEGA| x element of An i.o.)
=Intersection(for N=1 to infinity) U(n≥ N to infinity) (An)

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26
Q

limninf(A) =

A

={x element of OMEGA| x element of An e.v.}
=U(for N=1 to infinity) Intersection(n≥N to infinity) (An)

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27
Q

x element of An(i.o)

A

= for all N, element of natural numbers (there exists an n ≥N (x element of An))
= x belongs to infinitely many of the sets of An

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28
Q

x element of An(e.v.)

A

=there exists an N, element of natural numbers ( for all n≥N (x element of An))
= x belongs to every one of the sets of An from a specific n element of N

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29
Q

limninf(An) & limnsup(An) relationship:

A

1) limninf(An) subset limnsup(An)
2) limninf(An^c) = ( limnsup(An) )^c
3) ( limninf(An) )^c = limnsup(An^c)

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30
Q

What is a metric space (X,d)?

A

Exists from a non-empty set and a metric on that set

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31
Q

Properties of a metric space (X,d):

A

1) d(x,y) ≥ 0
2) d(x,y) = d(y,x)
3) d(x,y) = 0 <=> x=y
4) d(x,y) ≤ d(x,z) + d(z,y)

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32
Q

Given metric space (X,d), let (xn)n be a sequence of points in X, what is the definition of a limit of xn?

A

xn -> x if d(xn,x) -> 0

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33
Q

f is continuous:

A

=if f(xn) -> f(x) whenever xn -> x
=iff f^-1(A) is open whenever A is open
=iff f^-1(A) is closed whenever A is closed

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34
Q

Definition of an open set:

A

=Set is open if it contains NONE of A’s boundary points
=set A subset of X is open if for all x element of A, there exists an r >0 (B(x,r) subset of A)

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35
Q

Definition of a closed set:

A

=Set is closed whenever it contains ALL of A’s boundary points
=set A subset of X is closed if A^c = X-A is open

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36
Q

Definition of a closed set:

A

Set is closed whenever it contains ALL of A’s boundary points

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37
Q

Any union/finite intersection of open sets is?

A

Open

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38
Q

Any intersection/finite union of closed sets is?

A

Closed

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39
Q

Is the empty set open or closed?

A

Open and closed

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40
Q

Is this true: closed set = not open set

A

No

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41
Q

{x} open or closed?

A

closed for all x element of X

42
Q

Interior A^o =

A

= A - boundary(A)
=set of interior points of A

43
Q

Closure Ā =

A

= A or boundary(A)
= set of exterior points of A

44
Q

A^0 = A <=>

A

A is always open
~ (A n boundary(A)) = empty set

45
Q

Ā = A <=>

A

A is always closed
~ boundary(A) = subset of A

46
Q

Definition of a Cauchy sequence

A

= In metric space (X,d), xn is Cauchy if d(xn,xm) -> 0
as n -> infinity for n ≤m

= if xn -> x, xn is cauchy
~for complete spaces only

47
Q

Definition of Complete spaces:

A

Space (X,d) is complete if every Cauchy sequence is convergent

48
Q

1) Is real numbers^n complete for all n element of natural numbers?
2) Is real numbers complete?
3) Is rational numbers complete?

A

1) Yes
2) Yes
3) Not with the usual distance

49
Q

Definition of convergence:

A

sum of Un from n=1 to infinity = L

50
Q

Definition Absolute convergence:

A

If sum of |Un| from n=1 to infinity converges

51
Q

Relationship between convergence and absolute convergence:

A

If sum of |Un| from n=1 to infinity is absolutey convergent, then sum of Un from n=1 to infinity is convergent.

52
Q

Definition
of sigma-algebra:

A

Let C collection of subsets of Omega, then F = sigma(C)
=unique and smallest sigma-algebra that contains C
=F is generated by C
=F is generated by the smallest sigma-algebra

~sigma-algebra stable under countable applications of any set operations (union,intersection,difference, symmetric difference, liminf,limsup)

53
Q

Properties of sigma-algebra:

A

i) empty-set is element of F
ii) A element of F => A^c = omega \ A also an element of F
iii) If An element of F for n element of natural numbers, then U(n element of natural numbers) An element of F

~{countable union <=> sigma-algebra is countable}

54
Q

Definition of cardinality:

A

=|F|=2^# where # is the number of block partitions
=2^# possibilites
=size of respective sigma-algebra

55
Q

Definition of Block-partition:

A

A block partition of Omega is a set of subsets of Omega, {Bi: i element of I} such that:
i) U(i element of I)Bi = Omega &
ii) Bi n Bj = empty-set for i ≠ j

i: each Bi is a subset of Omega
ii: Bi’s are mutually exclusive

56
Q

Block-partition theorem 1:

A

If {Bi: i element of I} is a block-partition of Omega &
F = sigma({Bi:i element of I}) then any element A of F can be written as the disjoint union of any of the blocks.
A = Ú(i element of J)Bi, J subset of I

57
Q

Block-partition theorem 2:

A

If (Omega,F) is a measurable space & C = {Bn: n element of natural numbers} is a block partition of Omega, and F = sigma(C) then:

function f : (Omega,F) -> (Rbar, B(R)) is measurable iff f is constant in Bn for all n element of natural numbers

~f is measurable if Bn is constant within each block.

58
Q

Definition of Borel-sigma-algebra:

A

If Omega a topological space then:
B(R)
= sigma({open sets of Omega})
= sigma({closed sets of Omega})
= sigma({a,b: a,b elements of real numbers})

~essentially a sigma-algebra on R
~contains:
all open and closed sets

59
Q

Definition of Borel-sigma-algebra:

A

If Omega a topological space then:
B(Real numbers)
= sigma({open sets of Omega})
= sigma({closed sets of Omega})
= sigma({a,b: a,b elements of real numbers})

~essentially a sigma-algebra on |R
~contains:
all open and closed sets ;
all unions and intersections of open sets ;
all unions and intersections of closed sets

60
Q

Definition of Measurable space (Omega,F):

A

(Omega,F) is a measurable space if F is a sigma-algebra on a set Omega

~the elements of F =events/measurable sets

61
Q

Definition of Measure space (Omega,F,mu):

A

(Omega,F,mu) is a measure space if F is a sigma-algebra on the set Omega and mu is a measure on F

62
Q

Definition of Measure:

A

Given a measurable space (Omega,F), mu is a function mu: F -> complete |R is a measure iff:
i) 0 ≤ mu(A) < infinity (mu is non-negative)
ii) mu(empty set) = 0
iii) If A1,A2,… elements of F is a countable sequence of pairwise disjoint sets, then
mu(disjoint union of An) = sum(mu(An))

~mu is countably additive and non-negative
~ if mu(Omega) = 1, mu is a probability measure on the measure space (Omega,F,|P)

63
Q

Properties of a measure:

A

Suppose (Omega,F, mu) a measure space & A,B,A1,A2… elements of F

1) If A subset of B => mu(A) ≤ mu(B)

2) If A subset of B & mu(A) < infinity
=> mu(A-B) = mu(A) - mu(A n B)

3) If mu(A n B) < infinity
=> mu(A or B) = mu(A)+mu(B) - mu(AnB)

4) mu(union of An) ≤ sum(mu(An))

64
Q

When is a measure mu on (Omega,F) finite and sigma-finite:

A

finite: if mu(Omega) < infinity
sigma-finite:if Omega is a countable union of sets of finite measures

~e.g. lambda on (Real,Borel(Real))
lambda(Real) = infinity thus not finite
lambda( [n,n+1]) = 1 for all n thus is sigma-finite because each interval is one finite set.

65
Q

Proposition: Let C = {(-infinity, x]: x element of real numbers}, then Borel(Real numbers) = ?

A

Borel(Real) = sigma(C)

66
Q

Definition of Lebesgue measure:

A

Lebesgue measure on (Real, Borel(Real)) is the unique measure lambda on Real s.t.
lambda( (a,b) ) = lambda( [a,b] ) = lambda( (a,b] ) = b-a
= unique measure that assigns every interval its length

67
Q

Remark of Lebesgue: If A element Borel(Real) and A is countable then lambda(A)?

A

lambda(A)=0

68
Q

lambda(Q) =

A

0

69
Q

lambda(Z)=

A

0

70
Q

lambda(N)=

A

0

71
Q

lamba( {x} ) =

A

0

72
Q

Definition of Dirac Measure:

A

A measure on any (Omega,F) s.t. when we let x0 element of Omega, Sx0 is the dirac measure such that
Sx0 = 1 if x0 element of A & Sx0 = 0 if not

73
Q

Definition of Probability space:

A

(Omega, F, P) is a probability space if mu(Omega) = 1
Let A element of F, then
musubscriptA(B) = mu(A n B) = P(A n B) & musubscriptA(A) = P(A)

74
Q

In a probability space (Omega,F,P), what is PsubscriptA(B) ?

A

PsubA(B) = musubscriptA = P(A n B)/P(A)

75
Q

Continuity of measures: Given sequence of sets (An)n, An element of F (the sigma-algebra) for all n:
When is (An)n increasing ?
When is (An)n decreasing ?

A

(An)n increasing if A1 subset A2 subset A3 subset…
Thus An upwards arrow A
if A = union of An from 1 to infinity

(An)n decreasing if … subset A3 subset A2 subset A1
Thus An downwards arrow A
if A = intersection of An from 1 to infinity

76
Q

Proposition of continuity of measures:
i) If An upwards arrow A =>

ii) If An downwards arrow A and there exists n0 element of N s.t. mu(An0) < infinity =>

A

i) If An upwards arrow A => mu(An) ->mu(A)

ii) If An downwards arrow A and there exists n0 element of N s.t. mu(An0) < infinity => mu(An) -> mu(A)

77
Q

Fatou’s lemma:

A

Given A1,A2,… element of F, then
mu(liminf(An)) ≤ liminf(mu(An))
&
mu(limsup(An)) ≥ limsup(mu(An))

78
Q

Definition of a measurable function:

A

Given two measurable spaces
(Omega1, F1) and (Omega2, F2).

A function f: Omega1 -> Omega2 is measurable

if f^-1(B) element of F1 whenever B element of F2

thus f is F1/F2 measurable

79
Q

Indicator function as a special measurable function

A

Given A subset of Omega
IsubscriptA: (Omega,F) -> (Real,Borel(Real))
where
IsubscripA(w) = 1 if w element of A and 0 if not

80
Q

When is IsubscriptA measurable?

A

IsubscriptA <=> A element of F

~If IsubscriptA is measurable,
I^-1subscriptA(B) element of F for all B element of Borel(Real)

81
Q

Is a random variable X measurable?

A

Yes, X: (Omega,F,P) -> (Real,Borel(Real)

82
Q

Definition of Borel function:

A

If Omega a topological/measure space
then a measurable function
f: (Omega,B(Omega)) -> (R, B(Omega))

83
Q

If f: Omega->Real is continuous, what function is f?

A

Borel function

84
Q

Proposition 3.1.15 (measurable function f):
Suppose (Omega,F) is a measurable space and
let C = {Bn: n element of N} a block-partition of Omega and F = sigma(C), then?

A

f: (Omega,F) -> (Complete R,Borel(R)) is measurable
iff f is constant in Bn for all n element of N

f is measurable if Bn is constant in each block

85
Q

Definition of the following sigma-algebras generated by functions?

1) sigma(X)
2)sigma(X,Y)
3)When is Y X-measurable?

A

1) sigma(X)
=smallest sigma-alg that makes X measurable
=sigma( {X^-1(B) : B element of Borel(Real)

2)sigma(X,Y)
=smallest sigma-alg that makes X & Y measurable
= sigma( {X^-1(B), Y^-1(B): B element of Borel(Real) })

3)
<=> sigma(Y) subset sigma(X)

<=> sigma(X,Y) = sigma(X)

86
Q

What is limninf(An) in laymen’s terms?

A

All but finitely many of the events occur

87
Q

What is limnsup(An) in laymen’s terms?

A

Infinitely many of the events occur

88
Q

Is Z open or closed?

A

Closed

89
Q

Is Q open or closed?

A

Neither

90
Q

Is R open or closed?

A

Both

91
Q

Properties of measurable functions:

A

Suppose
f: (Omega,F) -> (Closure Real,B(Closure Real))
If f & g are measurable, then
1) f+g & f-g are measurable
2)f.g & f/g are measurable
3) a.f ( a = constant) is measurable

4) If fn : (Omega, F) -> Rbar is measurable for all n element of N and fn -> f pointwise
then f is measurable

~pointwise: (lim(n to infinity)fsubsriptn(w) = f(w) for all w element of Omega

92
Q

What space is the space of measurable functions?

A

A vector space

93
Q

Given a set S, what is the largest sigma-algebra on S?

A

Sigma-algebra F is a set of subsets of S, so F subset of P(S) (power set of S) given that the P(S) is a sigma-algebra.

Thus P(S) is the largest sigma-algebra on S.

94
Q

Given a set S, what is the smallest sigma-algebra on S?

A

F must contain ø and its compliment,S,
so {ø,S} subset of F.

Thus {ø,S} is the smallest sigma-algebra on S

95
Q

Given f: (Omega1,F1) ->(Omega2,F2) and g: (Omega2,F2) ->(Omega3,F3).
Is f,g and f o g measurable?

A

f is F1/F2-measurable
g is F2/F3-measurable
f o g is F1/F3-measurable

96
Q

Definition of sigma-algebra generated by a function:

A

Given f: Omega -> R
sigma(f) = ({ f^-1(B): B element of B(R) })

97
Q

Measurability theorem 1:
Given fn: (Omega,F) -> Closure R for all n element of N, then?

A

i) sup(fn) & inf(fn) is measurable
ii) limnsupn(fn) & limninfn(fn) is measurable
iii)limn(fn) (pointewise) is measurable

98
Q

Measurability theorem 2:
Given f,g: (Omega,F) -> Closure R are measurable, then
i) f v g = ?
ii) f n g =?
iii) f^+ = ?
iv) f^- = ?

A

i) f v g = max(f,g)
ii) f n g = min(f,g)
iii) f^+ = max(f,0)
iv) f^- = -min(f,0)

99
Q

Given Y: (Omega, sigma(Y)) ->
(Clo Real,Bore(Clo Real)).
Is Y measurable?

A

Yes,by definition.

100
Q

Given Y: (Omega, sigma(X)) ->
(Clo Real,Bore(Clo Real)).
Is Y measurable?

A

Yes, Y is X-measurable if you can find the find value of Y if we know the value of X.

101
Q

R = Rclo bar = ?

A

R bar = Rclo = [-infinity,infinity]

Financial Mathematics 378 (2 decks)

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