Sets, logic, measures & measurable functions Flashcards
Union: A or B
{x: x element of A or x element of B}
Intersection: A and B
={x: x element of A and x element of B}
Infinite union: U(Ai element of S}
={x: exists an i element of I (x element of Ai)}
Infinite intersection: N(Ai element of S)
={x: for all i element of I (x element of Ai)}
Set difference: A-B
=A - (A n B) = A n B^c
Symmetric Difference: A triangle B
=(A or B) - (A n B)
=(A-B) or (B-A)
=(A n B^c) or (A^c n B)
(A triangle B)^c
= A^c n B
Triangular inequality
=(A triangle C) subset (A triangle B) or (B triangle C)
A triangle C =
=(A triangle B) triangle (B triangle C)
When is a set A countable?
= 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
Push-forward function f: X -> Y, what is f(A)?
f(A) = {f(x) | x element of A}
~DONT preserve set operations
Push-back function f^-1: Y -> X, what is f^-1(A)?
f^-1(A) = {x element of A| f(x) element of B}
~DOES preserve set operations
max(A) = M <=>
for all x’s element of A (x ≤ M) and M element of A
min(A) = m <=>
for all x’s element of A (X ≥ m) and m element of A
Does min(A) and max(A) always exist?
No
sup(A) = M <=>
for all x’s element of A (x ≤ M) and M is the smallest of real numbers that satisfy this condition
inf(A) = m <=>
for all x’s element of A (x ≥ m) and m is the largest of real numbers that satisfy this condition
Does sup(A) and inf(A) always exist?
Yes
In laymen’s terms what is sup(A)?
its least UB
In laymen’s terms what is inf(A)?
its greatest LB
sup(empty set) =
infinity
inf(empty set) =
-infinity
If A is unbounded: sup(A) =
infinity
If A is unbounded: inf(A) =
-infinity
limnsup(An) =
={x element of OMEGA| x element of An i.o.)
=Intersection(for N=1 to infinity) U(n≥ N to infinity) (An)
limninf(A) =
={x element of OMEGA| x element of An e.v.}
=U(for N=1 to infinity) Intersection(n≥N to infinity) (An)
x element of An(i.o)
= 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
x element of An(e.v.)
=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
limninf(An) & limnsup(An) relationship:
1) limninf(An) subset limnsup(An)
2) limninf(An^c) = ( limnsup(An) )^c
3) ( limninf(An) )^c = limnsup(An^c)
What is a metric space (X,d)?
Exists from a non-empty set and a metric on that set
Properties of a metric space (X,d):
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)
Given metric space (X,d), let (xn)n be a sequence of points in X, what is the definition of a limit of xn?
xn -> x if d(xn,x) -> 0
f is continuous:
=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
Definition of an open set:
=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)
Definition of a closed set:
=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
Definition of a closed set:
Set is closed whenever it contains ALL of A’s boundary points
Any union/finite intersection of open sets is?
Open
Any intersection/finite union of closed sets is?
Closed
Is the empty set open or closed?
Open and closed
Is this true: closed set = not open set
No
{x} open or closed?
closed for all x element of X
Interior A^o =
= A - boundary(A)
=set of interior points of A
Closure Ā =
= A or boundary(A)
= set of exterior points of A
A^0 = A <=>
A is always open
~ (A n boundary(A)) = empty set
Ā = A <=>
A is always closed
~ boundary(A) = subset of A
Definition of a Cauchy sequence
= 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
Definition of Complete spaces:
Space (X,d) is complete if every Cauchy sequence is convergent
1) Is real numbers^n complete for all n element of natural numbers?
2) Is real numbers complete?
3) Is rational numbers complete?
1) Yes
2) Yes
3) Not with the usual distance
Definition of convergence:
sum of Un from n=1 to infinity = L
Definition Absolute convergence:
If sum of |Un| from n=1 to infinity converges
Relationship between convergence and absolute convergence:
If sum of |Un| from n=1 to infinity is absolutey convergent, then sum of Un from n=1 to infinity is convergent.
Definition
of sigma-algebra:
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)
Properties of sigma-algebra:
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}
Definition of cardinality:
=|F|=2^# where # is the number of block partitions
=2^# possibilites
=size of respective sigma-algebra
Definition of Block-partition:
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
Block-partition theorem 1:
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
Block-partition theorem 2:
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.
Definition of Borel-sigma-algebra:
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
Definition of Borel-sigma-algebra:
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
Definition of Measurable space (Omega,F):
(Omega,F) is a measurable space if F is a sigma-algebra on a set Omega
~the elements of F =events/measurable sets
Definition of Measure space (Omega,F,mu):
(Omega,F,mu) is a measure space if F is a sigma-algebra on the set Omega and mu is a measure on F
Definition of Measure:
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)
Properties of a measure:
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))
When is a measure mu on (Omega,F) finite and sigma-finite:
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.
Proposition: Let C = {(-infinity, x]: x element of real numbers}, then Borel(Real numbers) = ?
Borel(Real) = sigma(C)
Definition of Lebesgue measure:
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
Remark of Lebesgue: If A element Borel(Real) and A is countable then lambda(A)?
lambda(A)=0
lambda(Q) =
0
lambda(Z)=
0
lambda(N)=
0
lamba( {x} ) =
0
Definition of Dirac Measure:
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
Definition of Probability space:
(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)
In a probability space (Omega,F,P), what is PsubscriptA(B) ?
PsubA(B) = musubscriptA = P(A n B)/P(A)
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 ?
(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
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 =>
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)
Fatou’s lemma:
Given A1,A2,… element of F, then
mu(liminf(An)) ≤ liminf(mu(An))
&
mu(limsup(An)) ≥ limsup(mu(An))
Definition of a measurable function:
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
Indicator function as a special measurable function
Given A subset of Omega
IsubscriptA: (Omega,F) -> (Real,Borel(Real))
where
IsubscripA(w) = 1 if w element of A and 0 if not
When is IsubscriptA measurable?
IsubscriptA <=> A element of F
~If IsubscriptA is measurable,
I^-1subscriptA(B) element of F for all B element of Borel(Real)
Is a random variable X measurable?
Yes, X: (Omega,F,P) -> (Real,Borel(Real)
Definition of Borel function:
If Omega a topological/measure space
then a measurable function
f: (Omega,B(Omega)) -> (R, B(Omega))
If f: Omega->Real is continuous, what function is f?
Borel function
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?
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
Definition of the following sigma-algebras generated by functions?
1) sigma(X)
2)sigma(X,Y)
3)When is Y X-measurable?
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)
What is limninf(An) in laymen’s terms?
All but finitely many of the events occur
What is limnsup(An) in laymen’s terms?
Infinitely many of the events occur
Is Z open or closed?
Closed
Is Q open or closed?
Neither
Is R open or closed?
Both
Properties of measurable functions:
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
What space is the space of measurable functions?
A vector space
Given a set S, what is the largest sigma-algebra on S?
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.
Given a set S, what is the smallest sigma-algebra on S?
F must contain ø and its compliment,S,
so {ø,S} subset of F.
Thus {ø,S} is the smallest sigma-algebra on S
Given f: (Omega1,F1) ->(Omega2,F2) and g: (Omega2,F2) ->(Omega3,F3).
Is f,g and f o g measurable?
f is F1/F2-measurable
g is F2/F3-measurable
f o g is F1/F3-measurable
Definition of sigma-algebra generated by a function:
Given f: Omega -> R
sigma(f) = ({ f^-1(B): B element of B(R) })
Measurability theorem 1:
Given fn: (Omega,F) -> Closure R for all n element of N, then?
i) sup(fn) & inf(fn) is measurable
ii) limnsupn(fn) & limninfn(fn) is measurable
iii)limn(fn) (pointewise) is measurable
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^- = ?
i) f v g = max(f,g)
ii) f n g = min(f,g)
iii) f^+ = max(f,0)
iv) f^- = -min(f,0)
Given Y: (Omega, sigma(Y)) ->
(Clo Real,Bore(Clo Real)).
Is Y measurable?
Yes,by definition.
Given Y: (Omega, sigma(X)) ->
(Clo Real,Bore(Clo Real)).
Is Y measurable?
Yes, Y is X-measurable if you can find the find value of Y if we know the value of X.
R = Rclo bar = ?
R bar = Rclo = [-infinity,infinity]
