Integration, integrals, MCT & DCT

Question 1

Definition of an integral:

Answer 1

Given f: (Omega, F,mu) -> Rbar where f is measurable.
Integral of f w.r.t. measure mu:
Integral(f)dmu

Question 2

Integral w.r.t Dirac measure:

Answer 2

mu = Sx0 is a dirac measure.

Sx0(A) = {1, if x0 element of A & 0 otherwise

Integral(f)dSx0 = f(x0)

~if xi is in set A: if yes, Sxi(A)=1, if no Sxi(A) = 0

Question 3

General rule of integrating w.r.t. dirac measure:

Answer 3

If mu = a1Sx1 + a2Sx2+ …
where ai element of [0,infinity) & xi element of omega:

mu(A) = sum(for i)ai*Sxi(A)

Integral(f)dmu = sum(for i)f(xi)

Question 4

Integral w.r.t. lebesgue-measure:

Answer 4

f: (R,B(R),lambda)-> Rbar

Integral(f)dlamba = Integral(from -infinity to infinity) (f(x))dx

Question 5

Integral w.r.t. Lebesgue-Stieltjies measure:

Answer 5

If function F: Real->Real is
increasing (a ≤ b => F(a) ≤ F(b) ) &
right continuous (lim(x->a^+)F(x) = F(a)

musubscripF( (a,b)) = musubscripF([a,b]) = musubscripF((a,b]) = F(b)- F(a)

Question 6

If X: (Omega,F,Real) -> Real &
F(X) = P(X ≤ x), what is X’s distribution?

Answer 6

The Lebesgue-Stieltjies measure, musubscriptF(.).

Question 7

Integral w.r.t. pushed-measure:

Answer 7

If G: (Omega,F,Real) -> (S, L, muG^-1) & G is measurable.

(thus B element of L,G^-1(B) element of F because G is measurable)

muG^-1(B) = mu(G^-1(B))
(Push-measure = mu-measure of pullback)

muG^-1 is a measure on (S,L)

Question 8

If X: (Omega,F,Real) -> Real &
F(X) = P(X ≤ x), what is X?

Answer 8

A random variable

Question 9

True or False:
i) If a function G is measurable
<=> G is continuous?

ii) If a function G is continuous => G is measurable?

iii) If a function G is measurable => G is continuous?

Answer 9

i) False

ii) True

iii)False

Question 10

General rule of integrating a simple function:

Answer 10

Given f: (Omega,F,mu) ->(Rbar, B(Rbar) and assuming f is measurable.

If f = sum(for i)ai*Isubscript(Ai)
where ai element of Real &
Ai element of F, then

Integral(f)dmu = sum(for i)ai*mu(Ai)

Integral(Isubscript(A))dmu = mu(A)

Question 11

Definition of expectation:

Answer 11

Suppose X: (Omega,F,P) -> (Rbar,B(Rbar)) is measurable (thus X is measurable)
then
E[X] = Integral(X)dP

Question 12

Do integrals have to exist?

Answer 12

No

Question 13

Properties of integrating simple functions:

Answer 13

If f & g are simple functions , then

1) Integral(f)dmu doesn’t depend on the simple interpretation of f.

2) Integral(Isubscript(A))dmu = mu(A) for A element of F

3) Linearity:
If f,g ≥ 0 & a,b ≥ 0, then
Integral(af + bg)dmu = aIntegral(f)dmu +bIntegral(g)dmu

4)Monotonicity:

If f,g ≥ 0 & f ≤ g, then
Integral(f)dmu ≤ Integral(g)dmu

Question 14

What do we call measurable function whose integrals exist?

Answer 14

Integrable functions

Question 15

When is a function f: Omega -> Real
simple?

Answer 15

If it only takes a finite number of different values.

Question 16

If f: (Omega, F,mu) -> Clo Real, what is:
i) sF ?
ii) sF^+ ?
iii) mF ?
iv) mF^+ ?

Answer 16

i) sF : set of simple functions

ii) sF^+ : set of simple function ≥ 0

iii) mF : set of measurable functions

iv) mF^+ :set of measurable function ≥ 0

Question 17

If f: (Omega, F,mu) -> Rbar, what is:
i) sF ?
ii) sF^+ ?
iii) mF ?
iv) mF^+ ?

Answer 17

i) sF : set of simple functions

ii) sF^+ : set of simple function ≥ 0

iii) mF : set of measurable functions

iv) mF^+ :set of measurable function ≥ 0

Question 18

Integration for non-negative measurable functions, mF^+:

Answer 18

Given f: (Omega,F,mu) -> Rbar is measurable & f ≥ 0 then

Integral(f)dmu = sup{Integral(T)dmu:
T ≤ f & T element of sF^+}

Question 19

Integration steps:
Given f: (Omega, F,mu)->(Rbar,B(Rbar)) is measurable

Answer 19

1) If f is simple, i.e.,
f = sumofi(aiIsubscriptAi)
then
Integral(f)dmu = sumofi(aimu(Ai))

2) If f an element of mF^+ & f ≥ 0
then
Integral(f)dmu = sup{Integral(T)dmu:
T ≤ f & T element of sF^+}

Question 20

Integration for non-negative simple functions, sF^+:

Answer 20

If f simple,f = sumofi(aiIsubscriptAi), & f ≥ 0
then
Integral(f)dmu = sumofi(aimu(Ai))

Question 21

Facts about if f element of mF^+:

Answer 21

1) Integral(f)dmu always exists
~can be infinity

2) Integral(f)dmu ≥ 0

3) There exists Tn element of sF^+ s.t.:

(i) Tn is increasing (T1 ≤ T2 ≤ …)

(ii)Tn -> T as n -> infinity
(pointwise: lim(n to infinity)Tn(w) = f(w) , for all w element of Omega

(iii) Tn ≤ f for all n

then
Integral(f)dmu = lim(n to infinity)Integral(Tn)dmu

Question 22

Monotone convergence theorem:

Answer 22

Suppose fn: (Omega,F,mu) -> Rbar is measurable for all n element of N & f ≥0.

Assume that: lim n to infinity(fn) = f & f1 ≤ f2 ≤ …
(thus fn upwards arrow f)

then
lim n to infinity (Integral(fn)dmu)) = Integral(f)dmu

Question 23

MCT steps

Answer 23

Given fn(x) determine lim n to infinity Integral(fn)dmu

1) Find f(x) = lim n to infinity (fn(x))

2) Check if fn(x) is increasing as a sequence:
fn(x) ≤ fn+1(x) for all n

3) Use MCT results:

lim n to infinity (Integral(fn(x))dmu)) = Integral(f(x))dmu

Question 24

Integrating general functions:

Given f: (Omega,F,mu) -> Rbar is measurable
& recalling:
i) f^+ = max(x,0) ; f^- = -min(x,0)
thus f^+,f^- element of mF^+
ii) f = f^+ + f^-
what is Integral(f)dmu & when if f intergrable?

Answer 24

Integral(f)dmu = Integral(f^+)dmu + Integral(f^-)dmu
&
f is integrable if Integral(f)dmu exists & finite

Question 25

When is a function integrable?

Answer 25

If Integral(f)dmu exists and is finite.

Question 26

When is a function f element of mF mu-integrable?

Answer 26

IFF mu(|f|) = Integral(|f|)dmu < infinity OR IFF Integral(f^+)dmu < infinity & Integral(f^-)dmu < infinity

Question 27

Definition of mu-integrable functions, Ł^1(Omega,F,mu):

Answer 27

Ł^1(Omega,F,mu) = {f: Integral(|f|)dmu < infinity} and is a vector space.

Question 28

If f = g - h & g,h element of mF^+ then Integral(f)dmu=?

Answer 28

Integral(f)dmu = Integral(g)dmu - Integral(h)dmu

Question 29

True or false: Given Integral(f+g)dmu = Integral((f+g)^+)dmu - Integral((f+g)^-)dmu

Answer 29

True

Question 30

True or false: (f+g)^+ = f^+ + g^+

Answer 30

False

Question 31

Dominated Convergence theorem:

Answer 31

Let (fn)n be a sequence of integrable functions s.t. i)lim n to infinity fn(w) exists for all w element of Omega ii) There exists a function, g element of Ł^1(Omega,F,mu) s.t. |fn| ≤ g for all n element of N (thus its bounded by something independent of n) then lim n to infinity Integral(fn)dmu = Integral(lim n to infinity)(fn)dmu

Question 32

Given f is an integrable function and A is a measurable set, what is the integral of f over set A, IntegraloversetA(f)dmu =?

Answer 32

IntegraloversetA(f)dmu = Integral(f*IsubscriptA)dmu

Question 33

Fatou's lemma:

Answer 33

If fn element of mF^+ for all n element of N, then mu(liminf(fn)) ≤ liminf(mu(fn))

Question 34

Reverse Fatou's lemma:

Answer 34

If fn element of mF^+ fo all n element of N then limsup(mu(fn)) ≤ mu(limsup(fn))

Question 35

Fatou's lemma results ordered by size:

Answer 35

mu(liminf(fn)) ≤ liminf(mu(fn)) ≤ limsup(mu(fn)) ≤ mu(limsup(fn)) ~terms with limits on the inside is outside

Question 36

DCT Steps:

Answer 36

Given fn(x) determine lim n to infinity Integral(fn)dmu 1) Find lim n to infinity fn(x) ~get for different cases: x < 0; x = 0 ; x > 0 2) Find binding function g(x) for cases: x < 0; x = 0 ; x > 0 3) Integrate Integral(g(x))dmu 4) Use DCT results: lim n to infinity Integral(fn(x))dmu ≤ Integral ( lim n to infinity fn(x))dmu

Question 37

Given measurable function X: (Omega,F,P) -> (Rbar, B(Rbar), PX^-1) & X, what is: i) PX^-1? ii) E[X] = ?

Answer 37

i) PX^-1 : the distribution of X ii) E[X] = Integral over Rbar (x)dPX^-1(x) = Integral(X)dP

Question 38

Given X: (Omega,F,P) -> (Rbar,B(Rbar),PX^-1) & g: (Rbar,B(Rbar),PX^-1)-> (Rbar,B(Rbar),PX^-1) thus (Omega,F,P) -> (Rbar,B(Rbar),PX^-1) -> (Rbar,B(Rbar),PX^-1), change the variables of the integral & find E[g(X)]:

Answer 38

Change of variables: Integral(g)dPX^-1 = Integral(g o X)dP In general: E[g(X)] = Integral(g(x))dPX^-1(x)

Question 39

Inequality properties of Expectation:

Answer 39

1) Markov's inequality: If g ≥ 0 & increasing then E[g(X)] ≥ g(c)*P({X ≥ c}) for any c element of R 2) Chebyshev's inequality: For v > 0, then P({|X-E[X]|≥ v }) ≤ Var(X)/(v^2) where Var(X) = E[ (X-E[X])^2 ]

Question 40

If f is integrable, what is mu{f = ±infinity} ?

Answer 40

mu{f = ±infinity} = 0 thus f is finite mu-a.e.

Question 41

True or false: If f is integrable, |Integral(f)dmu| ≤ Integral(|f|)dmu

Answer 41

True

Question 42

Definition of changing the measure with density:

Answer 42

Given (Omega,F,mu) -> Rbar is measurable and f ≥ 0. Assume f element of Ł^1(Omega,F,mu), thus Integral(f)dmu < infinity Then define v = f*mu : F -> [0,infinity] s.t. v(A) = (f*mu)(A) = Integral over A(f)dmu = Integral(f*IsubscriptA)dmu. v is a measure on (Omega,F) & v <

Question 43

Chain rule:

Answer 43

Let v = f*mu & g: (Omega,F) -> Rbar s.t. g element of Ł^1(Omega,F,mu) then Integral(g)dv = Integral(gf)dmu since f = dv/dmu => dv = f*dmu

Question 44

Change of variables:

Answer 44

Given f: (Omega,F,mu) -> (T,S,mu(f^-1)) & g: (Rbar,B(Rbar),mu(f-1))-> Rbar & f,g are measurable. Then Integral(g)dmuf^-1 = Integral(g o f)dmu

Question 45

True or false: g o f = g(f)?

Answer 45

True

Question 46

Given (Omega,F,mu) is a measure space & A subset Omega: i) When is A a mu-null (a null set)? ii)When is the space (Omega,F,mu) complete? iii) What is the completion of F, Fbar?

Answer 46

i) If there exists a B element of F s.t. A subset B & mu(B)=0; ii) The space (Omega,F,mu) is complete if every mu-null set is in F. iii) Fbar = sigma( F or {A subset Omega: A is mu-null }) pg.8 notes L27-L29

Question 47

If (Omega, F, mu) is a measure space, when does a statement, I0(w), hold mu-almost everywhere (mu-a.e.)?

Answer 47

If { w element of Omega: IO(w) is false} is mu-null

Question 48

If (Omega, F, P) is a probability space, when does a statement, I0(w) is true P-almost surely (P-a.s.)?

Answer 48

If { w element of Omega: IO(w) is false} is P-null

Question 49

True or false: i) If F is complete, Fbar, then mu({w element of Omega: IO(w) false}) = 0 => IO(w) true a.e. ii) If F is complete, Fbar, then IO(w) true a.e. => mu({w element of Omega: IO(w) false}) = 0 iii) If F is complete, Fbar, then IO(w) true a.e. <=> mu({w element of Omega: IO(w) false}) = 0

Answer 49

i) TRUE ii) TRUE iii) TRUE

Question 50

Radon-Nikodym theorem:

Answer 50

If v & mu are sigma-finite measure on (Omega,F) & v< Rbar s.t. f = dv/dmu <=> dv = f.dmu f: density/Radon-Nikodym derivative

Question 51

Definition of a measure, v, being absolutely continuous w.r.t. another measure, mu (v<

Answer 51

A measure v on (Omega,F) is absolutely continuous w.r.t. mu if mu(A) = 0 => v(A) =0; thus v<

Question 52

Using Radon-Nikodym theorem:

Answer 52

Find f(x) = lim h to o^+ (mu([x, x+h])/(lambda([x, x+h])

Question 53

Definition of product of measure spaces:

Answer 53

Let (Omega1,F1) & (Omega2,F2) be measurable spaces. Define projections maps: pie1 = Omega1 x Omega2 -> Omega1 ( (w1,w2) -> pie1(w1,w2) = w1 ) & pie2 = Omega1 x Omega2 -> Omega2 ( (w1,w2) -> pie2(w1,w2) = w2 ) Define F1 circle* F2 = sigma(pie1,pie2) to be the smallest sigma-algebra that makes pie1 & pie2 measurable. ~(w1,w2): sample point in a space of combined outcomes (w1 element of Omega1 occurred & w2 element of Omega2 occurred) ~thus given combined outcome (w1,w2), pie1(w1,w2)=w1 measures which outcome occurred in Omega1 & pie2(w1,w2)=w2 measures which outcome occurred in Omega2 ~projection maps pie1 & pie2 are measurable given that if we know a combined outcome (w1,w2), we should know the component outcomes, w1 & w2.

Question 54

True or false: B(R) circlex B(R) = B(R^2)

Answer 54

True

Question 55

Definition of product of measures:

Answer 55

Let (S,F1,mu1) & (T,F2,mu2) be sigma-finite measure spaces. Define map mu1 circlex mu2: F1 circlex F2 ->Rbar^+ Then product measure of mu1 & mu2 on (S circlex T, F1 circlex F2) for B element of F1 circlex F2: mu1 circlex mu2 = Int(Int (IsubB(s,t)) mu2(dt))mu1(ds)

Question 56

Fubini's theorem:

Answer 56

Assume (Omega1,F1,mu1) & (Omega2,F2,mu2) are sigma-finite measure spaces then if f element of Ł^1(Omega1 x Omega2, F1 cricx F2, mu1 circx mu2) (mu1 circx mu2)(f) = Int(f)dmu1circx mu2 = Int( Int(f(x,y)dmu1(x)) dmu2(y) =Int( Int(f(x,y)dmu2(y)) dmu1(x) ~ Int( Int(f)dv)du = Int( Int(f)du)dv ~f element of Ł^1 <=> abs. conv. Int(|f|)dmu1dmu2 < inf

Question 57

Jensen's inequality for E[X]:

Answer 57

If g: R->R is convex (function as a + 1st derivative: bend upwards), then E[g(X)] ≥ g(E[X])

Question 58

Properties of expectation:

Answer 58

1) If X ≤ Y a.s. then E[X] ≤ E[Y] 2) E[aX+bY] = aE[X] + bE[Y] 3) If X = Y a.s. then E[X]=E[Y] 4) Continuity: Assume X -> X a.s. , then (i) Xn≥ 0 & Xn upwardsarrow X a.s. (non-neg & increasing) then E[Xn] -> E[X] (ii) There exists Y s.t. E[Y] < inf & |Xn| ≤ Y a.s. then E[Xn] -> E[X] (iii) & Xn is bounded, exists a constant c s.t. |Xn| ≤ c then E[Xn] -> E[X] & E[Xn^k] -> E[X^k] for k element of N 5) E[c] = c if c a constant

Question 59

Definition of correlation:

Answer 59

Let X,Y element of Ł^1(omega,F,P) then correlation px,y = cov(X,Y)/(sd(X)*sd(Y)) & -1 ≤ px,y ≤ 1 ~X,Y are correlated if cov(X,Y) = 0 or px,y = 0.

Question 60

Definition of covariance:

Answer 60

Let X,Y element of Ł^1(omega,F,P) then covariance, cov(X,Y) = E[ (X-E[X]) (Y-E[Y]) ) = E[XY]-E[X]E[Y]

Question 61

True or false: Uncorrelated ≠ Independent

Answer 61

True

Question 62

True or false: Cov & correlation measures only LINEAR dependence (Y = aX+b)

Answer 62

True

Financial Mathematics 378 (2 decks)

• Sets, logic, measures & measurable functions
• Integration, integrals, MCT & DCT