Lebesgue

Question 1

Let S be a set. The power set of S, P(S) is the:

Answer 1

collection of all subsets of S. If S has finite cardinality, then |P(S)|=2^|S|

Question 2

Let Å⊆P(S). Å is called a sigma field over S if:

Answer 2

1: ∅ ∈ Å.
2: if A ∈ Å, then S\A ∈ Å (complement of A also in Å)
3: if A1, A2,… is a sequence of elements of Å, then also the union of Ak is an element of Å

Question 3

If Å is a sigma field, then it can be used as a domain for a measure, and sigma fields are closed under complements, which means:

Answer 3

that whenever you take one or more sets from your collection and perform an operation, the resulting set is still in the collection

Question 4

Let S be a set and let G⊂P(S). Define σ(G)={B ∈ P(S)|B ∈ Å for every sigma field Å containing G}. Then σ(G) is:

Answer 4

a sigma field, and it is the smallest one that contains G

Question 5

Let G1 and G2 be two collections of subsets of S, so G1,G2⊂P(S). If G1⊆σ(G2), then:

Answer 5

σ(G1)⊆σ(G2)

Question 6

Let S⊆R be an interval and let G1={(a,b]|a,b ∈ S, a<b}. Then σ(G1) is called:

Answer 6

the Borel sigma field of S, denoted β_s, it is the smallest sigma-field that contains all intervals of the form (a,b]. It basically contains all the usual subsets of R that are encountered in analysis

Question 7

A measure is:

Answer 7

a function that assigns to every element of a sigma field a size

Question 8

Let Å be a sigma field over S. A function μ: Å–>[0,∞] is called a measure if:

Answer 8

1: μ(∅)=0
2: μ is σ-additive (the measure of the union of all Ak = Σmeasures of all Ak) if all Ak’s are elements of Å and are pairwise disjoint

Question 9

A measurable space is:

Answer 9

a pair (S,Å) in which S is a set and Å is a sigma field over S

Question 10

A measured space is:

Answer 10

a triple (S,Å,μ) in which (S,Å) is a measurable space and μ is a measure over Å. Elements of Å are called Å-measurable sets.

Question 11

A counting measure is:

Answer 11

a set S, with Å its powerset P(S), where t(A) = number of elements of A

Question 12

A probability measure is:

Answer 12

a set S, the sample space of an experiment, then μ(A) denotes the probability that the outcome of the experiment is somewhere in A

Question 13

Let (S,Å,μ) be a measured space, then the following properties hold:

Answer 13

1: Let A, B ∈ Å with A⊆B. Then μ(B)=μ(A)+μ(B\A), thus μ(A)<=μ(B)
2: If A1⊆A2⊆… is an increasing sequence of sets in Å, then we have: the measure of the union of all Ak = the limit of the measure of Ak, with k –> ∞
3: If B∞⊆,…,⊆B2⊆B1 is a decreasing sequence of sets in Å, then we have: the measure of the intersection of all Bn = the limit of the measure of Bn, with n –> ∞

Question 14

1: λ((-∞,b])
2: λ((a,b))
3: λ({b})
4: λ(Z)

Answer 14

1: use that (-∞,b]=union(-n,b] = ∞, where n–>∞
2: use that (a,b)=union(a,b-1/n] = b-a , where n–>∞
3: use that (a,b]=(a,b)U{b} = 0
4: use that Z is countable = 0

Question 15

Let (S,Å) be a measurable space. A function f:S–>RU{-∞,∞} is called Å-measurable if:

Answer 15

{s∈S|f(s)<=a} ∈ Å for each a ∈ RU{-∞,∞}

Question 16

1: f v g
2:f ^ g
3: f+
4: f-
5: |f|
6: (f v g) + (f ^ g)
7: f+ - f-
8: f+ + f-

Answer 16

1: max{f,g}
2: min{f,g}
3: max{f,0}
4: -min{f,0}
5: absolute value of f
6: f + g
7: f
8: |f|

Question 17

A function f: S–>R+ is called simple, denoted by T+ if:

Answer 17

1: f≥0
2: f is Å-measurable
3: f assigns only finitely many different values

Question 18

Let f ∈ T+ and suppose that f attains n different positive values a1,a2,…,an. Define for all i in {1,….,n}, Ai to be the inverse image of ai:

Answer 18

Ai=f^-1(ai)={s∈S|f(s)=ai}. Then Ai ∈ Å, the sets Ai are pairwise disjoint and for all x ∈ S, we have the standard form: f= Σai*1(A1)

Question 19

Let (S,Å,μ) be a measured space. If f ∈ T+ has the standard form, then its Lebesgue integral is defined by:

Answer 19

∫fdμ=Σai*μ(Ai)
This integral is well defined and can be infinite

Question 20

Let f1,f2,… be a sequence of Å-measurable functions with shared domain S. Then fn↗f denotes monotone convergence:

Answer 20

the sequence of functions is increasing, f1(x)≤f2(x)≤… for all x in S, and has pointwise limit f

Question 21

If f is Å-measurable and non-negative, then:
Note: the class of non-negative Å-measurable functions is denoted by I+

Answer 21

fn↗f for some sequence of simple functions f1,f2,…

Question 22

Let (S,Å,μ) be a measured space. If f∈I+, then:

Answer 22

∫fdμ=sup{∫gdμ|g∈T+,g≤f}, which can be interpreted as the smallest upper bound of all underestimations

Question 23

Let (S,Å,μ) be a measured space. Let f1,f2,… ∈ I+ and fk↗f. Then:
Monotone Convergence Theorem (MCT)

Answer 23

f ∈ I+ and ∫fkdμ–>∫fdμ, when k–>∞

Question 24

Let (S,Å,μ) be a measured space. Then for every f,g∈ I+:

Answer 24

f+g ∈ I+ and ∫(f+g)dμ = ∫fdμ + ∫gdμ

Question 25

Let (S,Å,μ) be a measured space. An Å-measurable function f:S-->RU{-∞,∞} is called Lebesgue integrable if:

Answer 25

at least one of the functions f+ and f- in I+ has a finite Lebesgue integral: ∫f+dμ<∞ or ∫f-dμ<∞. Then its Lebesgue integral is: ∫fdμ=∫f+dμ - ∫f-dμ. The class of Lebesgue integrals is defined by I

Question 26

(Lebesgue's dominated convergence theorem): Let (S,Å,μ) be a measured space. let f1,f2,...:S-->RU{-∞,∞} be a sequence of Å-measurable functions that pointwise converges to a function f:S-->RU{-∞,∞}. Assume the existence of a function g in I+ such that ∫gdμ<∞ and |fn|≤g for every n. Then:

Answer 26

f1,f2,... ∈ I and ∫fndμ-->∫fdμ, where n-->∞

Question 27

Let (S,Å,μ) be a measured space. Let f ∈ I+ and A={x∈S|f(x)>0}. Then:

Answer 27

∫fdμ=0 if and only if μ(A)=0

Question 28

Let f:[a,b]-->R be Riemann integrable. Then:

Answer 28

f is Lebesgue integrable as well with respect to the Borel-Lebesgue measure λ and ∫fdλ=∫f(x)dx

Question 29

Let f:[a,b]-->R be Stieltjes integrable with respect to the right-continuous integrator g. Then:

Answer 29

f is Lebesgue integrable with respect to λ_g. Furthermore: ∫fdλ_g=∫fdg

Question 30

Let X,X1,X2,... be a sequence of non-negative random variables such that P(Xk≤X(k+1) for all k ∈ N and limXk=X) = 1. Then:

Answer 30

limEXk=EX

Question 31

Let X,X1,X2,... be a sequence of random variables satisfying P(limXk=X) = 1. If there exists a random variable Y with finite mean such that P(|Xk|≤Y) = 1 for all k ∈ N, then:

Answer 31

the expectations EX and EXk exist, are finite, and limEXk=EX

Question 32

If {t} ∈ C, then 1_{t} is:

Answer 32

a C-measurable function

Question 33

If A=P(S), then every function on S is:

Answer 33

measurable

Question 34

Let S⊆R be an interval, possibly unbounded. Then every increasing function f: S-->R is:

Answer 34

Borel-measurable