Sets, logic, measures & measurable functions

Question 1

Union: A or B

Answer 1

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

Question 2

Intersection: A and B

Answer 2

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

Question 3

Infinite union: U(Ai element of S}

Answer 3

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

Question 4

Infinite intersection: N(Ai element of S)

Answer 4

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

Question 5

Set difference: A-B

Answer 5

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

Question 6

Symmetric Difference: A triangle B

Answer 6

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

Question 7

(A triangle B)^c

Answer 7

= A^c n B

Question 8

Triangular inequality

Answer 8

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

Question 9

A triangle C =

Answer 9

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

Question 10

When is a set A countable?

Answer 10

= 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

Question 11

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

Answer 11

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

Question 12

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

Answer 12

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

Question 13

max(A) = M <=>

Answer 13

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

Question 14

min(A) = m <=>

Answer 14

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

Question 15

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

Answer 15

No

Question 16

sup(A) = M <=>

Answer 16

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

Question 17

inf(A) = m <=>

Answer 17

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

Question 18

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

Answer 18

Yes

Question 19

In laymen’s terms what is sup(A)?

Answer 19

its least UB

Question 20

In laymen’s terms what is inf(A)?

Answer 20

its greatest LB

Question 21

sup(empty set) =

Answer 21

infinity

Question 22

inf(empty set) =

Answer 22

-infinity

Question 23

If A is unbounded: sup(A) =

Answer 23

infinity

Question 24

If A is unbounded: inf(A) =

Answer 24

-infinity

Question 25

limnsup(An) =

Answer 25

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

Question 26

limninf(A) =

Answer 26

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

Question 27

x element of An(i.o)

Answer 27

= 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

Question 28

x element of An(e.v.)

Answer 28

=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

Question 29

limninf(An) & limnsup(An) relationship:

Answer 29

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

Question 30

What is a metric space (X,d)?

Answer 30

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

Question 31

Properties of a metric space (X,d):

Answer 31

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)

Question 32

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

Answer 32

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

Question 33

f is continuous:

Answer 33

=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

Question 34

Definition of an open set:

Answer 34

=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)

Question 35

Definition of a closed set:

Answer 35

=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

Question 36

Definition of a closed set:

Answer 36

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

Question 37

Any union/finite intersection of open sets is?

Answer 37

Open

Question 38

Any intersection/finite union of closed sets is?

Answer 38

Closed

Question 39

Is the empty set open or closed?

Answer 39

Open and closed

Question 40

Is this true: closed set = not open set

Answer 40

No

Question 41

{x} open or closed?

Answer 41

closed for all x element of X

Question 42

Interior A^o =

Answer 42

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

Question 43

Closure Ā =

Answer 43

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

Question 44

A^0 = A <=>

Answer 44

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

Question 45

Ā = A <=>

Answer 45

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

Question 46

Definition of a Cauchy sequence

Answer 46

= 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

Question 47

Definition of Complete spaces:

Answer 47

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

Question 48

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

Answer 48

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

Question 49

Definition of convergence:

Answer 49

sum of Un from n=1 to infinity = L

Question 50

Definition Absolute convergence:

Answer 50

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

Question 51

Relationship between convergence and absolute convergence:

Answer 51

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

Question 52

Definition of sigma-algebra:

Answer 52

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)

Question 53

Properties of sigma-algebra:

Answer 53

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}

Question 54

Definition of cardinality:

Answer 54

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

Question 55

Definition of Block-partition:

Answer 55

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

Question 56

Block-partition theorem 1:

Answer 56

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

Question 57

Block-partition theorem 2:

Answer 57

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.

Question 58

Definition of Borel-sigma-algebra:

Answer 58

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

Question 59

Definition of Borel-sigma-algebra:

Answer 59

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

Question 60

Definition of Measurable space (Omega,F):

Answer 60

(Omega,F) is a measurable space if F is a sigma-algebra on a set Omega ~the elements of F =events/measurable sets

Question 61

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

Answer 61

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

Question 62

Definition of Measure:

Answer 62

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)

Question 63

Properties of a measure:

Answer 63

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))

Question 64

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

Answer 64

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.

Question 65

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

Answer 65

Borel(Real) = sigma(C)

Question 66

Definition of Lebesgue measure:

Answer 66

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

Question 67

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

Answer 67

lambda(A)=0

Question 68

lambda(Q) =

Answer 68

0

Question 69

lambda(Z)=

Answer 69

0

Question 70

lambda(N)=

Answer 70

0

Question 71

lamba( {x} ) =

Answer 71

0

Question 72

Definition of Dirac Measure:

Answer 72

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

Question 73

Definition of Probability space:

Answer 73

(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)

Question 74

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

Answer 74

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

Question 75

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 ?

Answer 75

(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

Question 76

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 =>

Answer 76

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)

Question 77

Fatou's lemma:

Answer 77

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

Question 78

Definition of a measurable function:

Answer 78

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

Question 79

Indicator function as a special measurable function

Answer 79

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

Question 80

When is IsubscriptA measurable?

Answer 80

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

Question 81

Is a random variable X measurable?

Answer 81

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

Question 82

Definition of Borel function:

Answer 82

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

Question 83

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

Answer 83

Borel function

Question 84

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?

Answer 84

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

Question 85

Definition of the following sigma-algebras generated by functions? 1) sigma(X) 2)sigma(X,Y) 3)When is Y X-measurable?

Answer 85

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)

Question 86

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

Answer 86

All but finitely many of the events occur

Question 87

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

Answer 87

Infinitely many of the events occur

Question 88

Is Z open or closed?

Answer 88

Closed

Question 89

Is Q open or closed?

Answer 89

Neither

Question 90

Is R open or closed?

Answer 90

Both

Question 91

Properties of measurable functions:

Answer 91

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

Question 92

What space is the space of measurable functions?

Answer 92

A vector space

Question 93

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

Answer 93

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.

Question 94

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

Answer 94

F must contain ø and its compliment,S, so {ø,S} subset of F. Thus {ø,S} is the smallest sigma-algebra on S

Question 95

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

Answer 95

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

Question 96

Definition of sigma-algebra generated by a function:

Answer 96

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

Question 97

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

Answer 97

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

Question 98

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^- = ?

Answer 98

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

Question 99

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

Answer 99

Yes,by definition.

Question 100

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

Answer 100

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

Question 101

R = Rclo bar = ?

Answer 101

R bar = Rclo = [-infinity,infinity]