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Math BSc 4th Sem > Measure and Integration > Flashcards

Measure and Integration Flashcards

1
Q

1 Subsection I.1 and Theorem I.1.8
Define (sig)ring, (sig)algebra and Dynkin
If E s.s.o. 2^Omega is n-closed, then d(E)=sig(E)
semiring, semifield/semialgebra
Where does the latter occur?

A
  1. pg 9
    pg 16
    Prop 1.9
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2
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2 Set functions, measures up to Propositino I.2.4
Define :
content, premeasure, measure
E.g. 1) of a simple probability measure,
2) an additive measure which is not sig-additive,
3) (what about the sum of a sequence of measure?) under which condition does a linear combination of a sequence of probability measures give another probability measure?
4) counting measure?
5) Can we define the lebesgue measure?
6) What about measures with functions?

A

1) dirac is a probability measure
2) Omega=N, R={A in P(Omega) s.t. A or Omega\A is finite}, mü(A):= 0 if finite, :=inf if Omega\A finite
3) (all three form a convex cone) when the sum of the coefficients equals 1,
4) mü(A)=|A| when A finite, else :=inf, is a content, premeasure if R ring, measure if R sig-algebra,
5) No for that we need Carathéodory, however we can define (a,b]:={x in R^d : ai

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3
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3 Theorem I.2.6
State the theorem of equivalences for premeasures,
Where is this used?

A
  1. pg 14
    With the help of compact classes we use 5’) to show show sig-additivity in the Theorem of Daniell-Kolmogorov, and use 5’) to show sig-additivity in the Theorem of Ionescu-Tuclea
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4
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4 Theorem I.2.9 and Proposition I.2.12
Thm: How does the ring generated by a semiring look like?
Prop: What can we say about a additive function mü on a semiring?

A
  1. pg. 16
    Thm: What needs to be proven?
    Prop: It uniquely extends to a content on R,
    Pf: take two representations
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5
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5 Theorem I.2.13
State the conditions under which a topological space with a ring and a finite content gives rise to a pre-measure.
Outline the proof.

A
  1. pg. 18
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6
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6 Proposition I.2.17
Define distribution function.
State under which conditions and between which distribution functions and which set-functions there exists a bijection.
Pf: Outline

A
  1. pg 21
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7
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7 Definition I.3.1 and the fact that with a measure one can define an outer measure
Define outer measure, measurable set,
give an outer measure induced by a measure (both equivalent constructions)
Skip to sub-additivity of last.

A
  1. pg 23-24

show the last is a measure and equivalent

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8
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8 Theorem I.3.3 and the first part of the proof
State Theorem of Carathéodory
Outline proof.

A
  1. pg 24
    1) extend mü to outer-measure
    2) For every outer measure ,mü, the family A of mü-measurable sets is a sig-field and mü is a measure on it
    3) R s.s.o. A* and mü=mü on R.
    4) restrict mü     to mü~ and uniqueness (in sep prop)
    Outline of 1):
    Def U(C):={ (An)n : An in R and C s.s.o. Union(An)}
    Def mü(C):=inf{Sum(mü(An) : (An)n in U(C)}
    mü*({})=0 and mon easy. For sig-subadd prop, choose for (Cn)n sequences (Akn)k in R from U(Cn) with Sum(mü(Akn))=
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9
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9 Theorem I.3.3 and the second part of the proof

State which result is proven in this part.

A
  1. pg. 25
    2) For every outer measure ,mü, the family A of mü-measurable sets is a sig-field and mü is a measure on it.

Show A* is a field, for union expand set measurable equation and replace C by Cn(AuB)….

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10
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10 Theorem I.3.3 and the third and the fourth part of the proof
Outline 3) and 4) and state Proposition used in 4).

A
  1. pg. 26
    3) R s.s.o. A* and mü=mü on R.
    4) restrict mü     to mü~ and uniqueness (in sep prop)
    Prop: Let E s.s.o. 2^Omega be n-closed and A=sig(E). Suppose there exists a sequence (Ei)i in E with Union(Ei)=Omega. If mü1, mü2 are measures on A with mü1=mü2 on E and mü1(Ei)=mü2(Ei)
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11
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11 Definition I.3.5 and Lemma I.3.6
Define complete measure space.
State the lemma about the completeness and Carathéodory.

A
  1. pg. 27
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12
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12 Theorem I.3.7

State the existence theorem (and definition) of the completion of a measure space.

A

12 pg. 28

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13
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13 Theorem I.3.8

State the theorem relating completion and the Carathéodory theorem

A

13 pg. 29

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14
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14 Example I.3.9 (Cantor)
In the nth step of construction, what does Cn look like?
Topological properties?
Define the devil’s staircase
Prove continuity
Can we extend this function onto all of R?

A
  1. 2^n closed intervals Cni (i in {1,…,2^n) each of length 3^-n.
    Cpt, it is closed and bounded [0,1] (Heine-Borel).
    Uncountable:
    We write all numbers in triadic repr. x=Sum(xk3^-k: xk in {0,1,2}), making this representation unique by demanding that xk != 0 for at most finitely many k (i.e. always representation with less zeros), then phi(x):=Sum(phi(xk)2^k) is surjective with phi:C–>[0,1] (by dyadic representation of numbers in [0,1]).
    Nowhere dense in [0,1]: C is its own closure (for it is closed) and has empty interior, so it is nowhere dense OR it has no accumulation points.
    Distribution function: pg. 32
    Continuity: ‘’
    Yes, we can extend this to a continuous function because F is continuous and defined on a dense set R\C.
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15
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15 Lebesgue measure and Lemma I.4.4
On what is the lebesgue measure defined?
Translation invariance of Lebesgue measure: outline proof.

A
  1. It is first defined as a content on R:={(a,b] : a,b in R^d} and we had a proposition showing that contents on a semiring uniquely extend to a pre-measure on the generated ring, which we call the Lebesgue pre-measure. From there we use Carathéodory to show that the Lebesgue pre-measure uniquely extends to a measure on (what turns out to be by last Prop) the Borel sig-algebra, which we call the Lebesgue measure. (Note however, that A=sig(R)=B(R^d) != A*=A^v != 2^R^d).
    pg 34
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16
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16 Theorem I.4.7 on non-Lebesgue measurable sets
State the theorem and the lemmas it uses.
Outline: proof (state how second assertion follow from first)
Where did we use the axiom of choice and can the use of the axiom of choice be avoided?

A
  1. Lem 4.5: If A is Borel and lam(A)>0, there exists for every alpha in (0,1) an interval J_alpha with lam(A n J_alpha)>alphalam(J_alpha)>0.
    Lem 4.6: If xi in R\Q, then the sets:
    Q_a:=Z+xi    Z, Q_o:=Z+2xiZ and Q_o:=Z+xi*(2Z+1) are all dense in R.
    Thm: There exists a set M in 2^R s.t.
    (4.1) lam(B)=0 for all B Borel s.s.o. M,
    (4.2) lam(B)=0 for all B Borel s.s.o. R\M.
    In particular M is non lebesgue-measurable.
    No, in the choice of representatives.
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17
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17 Theorem II.1.14
Prove for (fn)n measurable, that sup fn, inf fn, limsup fn, liminf fn are all measurable.
A

17 pg. 49

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18
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18 Theorem II.1.18
Let Omega arbitrary but non-empty, (Omega’,A’) a measurable space, phi:Omega–>Omega’ a mapping and f:Omega–>R a function. Then f is sig(phi)-B(R)-measurable if and only if there exists an A’-measurable function g:Omega’–>R s.t. f=g o phi.

A
  1. pg. 51
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19
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19 Lemma II.2.2 and Definition II.2.3 and Corollary II.2.4
Lem 2.2: Show that if f in E_+ has two representations f=Sum(aiI_Ai)=Sum(biI_Bi), then I(f)=Sum(aimü(Ai))=Sum(bimü(Bi)).
Define the integral for a step function.
Show additivity and monotonicity of the integral of step functions.

A

19 pg.53

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20
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20 Lemma II.2.6

Express monotone convergence for E_+ functions.

A

20 pg. 55

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21
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21 Proposition II.2.7 and Definition II.2.8
State: Prop 2.7: Uniqueness of limits integrals of E_+ functions converging towards a function.
Define integral of functions in E*+. (What is E*+?)
How can we characterize this def?

A

21 pg. 56

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22
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22 Theorem II.2.9 and Corollary II.2.10
State monotone convergence, Beppo-Levi
And its corollaries concerning series.
Outline proof.

A
  1. pg. 58
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23
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23 Proposition II.2.14
Let f,g be in bar(L)^1, a in R.
Prove: Linearity, Additivity (when well-defined), Monotonicity, Triangle inequality, max(f,g)/min(f,g) in bar(L)^1.

A
  1. pg. 60
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24
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24 Lemma II.2.17:

Suppose that f in bar(L)^1(mü) or f in E_+ with I(f)

A
  1. pg. 62
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25
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25 Theorem II.3.1 (monotone convergence)
State theorem,
name counter example when condition g is dropped, prove, what if g condition only mü-a.e.?
Hint: fn something g –>

A
  1. pg. 65, counterexample: indicator fn := 1_n,n+1
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26
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26 Theorem II.3.2 (Fatou’s Lemma)
State theorem, name counter example when condition g is dropped (because of following comment), prove, what if g condition only mü-a.e.?
Hint 4 pf: hm:=inf (fn : n>=m)

A
  1. pg. 66, counterexample: indicator fn := 1n,n+1 (sup form, else -fn for inf form) https://math.stackexchange.com/questions/2051/fatous-lemma-counterexample
    Is e.g. 3.9,1) fn:=n*1    (0,1/n) also an example?
27
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27 Theorem II.3.4 (dominated convergence, Lebesgue) State theorem
Hint: Fatou

A
  1. pg. 66
28
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28 Definition II.3.5 and Lemma II.3.7:
Define converges mü-almost everywhere, mü-stochastically and in L^1(mü) (stiff L),
state Markov’s inequality and prove it.
Hint: Indicator function using f and h.

A
  1. pg. 67-68
29
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29 Theorem II.3.8: Convergence mü-almost everywhere and convergence in L^1(mü) both imply mü-stochastic convergence.
Name an example of a sequence that converges mü-stochastically but has a different limit in L^1(mü).
Name an example of a sequence that converges mü-stochastically but not mü-almost everywhere.
Prove the theorem.

A
  1. Example 3.9, 1) fn:=nI_(0,1/n).
    Example 3.9, 2) fn:= I_[(k-1)    2^-n,k*2^-n].
    pg. 68
30
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30 Theorem II.3.10: Suppose that (Omega, A, mü) is sigma-finite. Then we have fn–>f mü-stochastically if and only if every subsequence of (fn) contains a further subsequence which converges to f mü-almost everywhere.

A
  1. pg. 69
31
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31 Remark II.3.11

Do exercises in text

A

31 pg. 70

32
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32 Example II.3.12:
Show that the conditions in DCT are not necessary
Definition II.3.13: define uniformly (mü-)integrable
Remark II.3.14: Explain the different characterisations of uniform integrability (general and finite case)

A

32 pg. 71

33
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33 Remark II.3.16:
Show that a uniformly integrable family of functions is bounded.
Lemma II.3.17:
State the “Is this surprising” lemma, that looks like continuity.
Hint for proof: choose: delta=eps/(2c)

A

33 pg. 73

34
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34 Theorem II.3.18

For (fn)n and f in bar(L)^1(mü) what is fn–> f in L^1 equivalent to?

A

34 pg. 74

35
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35 Proposition II.3.21
Suppose that fn:Omega fn:Omega–>bar(R) are measurable and the family (f^-)n is uniformly mü-integrable. Then I(liminf fn)=

A

35 pg. 76

36
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36 Theorem II.3.22 Egorov

State the theorem and prove it.

A
  1. pg. 77
37
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37 Theorem II.4.2 Hölder and Minowski inequalities. State and prove them.

A
  1. pg. 79
38
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38 Proposition II.4.4 State and prove the theorem about L^p spaces monotonously decreasing under a condition
Hint: f=f1 with g=1, C:=||1||

A
  1. pg. 81
39
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39 Theorem II.4.11
State Fischer-Riesz
Hint: What case distinction do we make?

A

39 pg. 83

40
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40 Define the essential supremum.
Theorem II.4.14
Under which assumption does the ess sup exist? Is is unique? What holds for (fn)n?

A

40 pg. 85

41
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41 Theorem II.5.3
Define signed measure.
State the Theorem on the Hahn decomposition.

A
  1. pg. 87
42
Q

42 State the two lemmas (/def) about mü_f.

Lemma II.6.1 and Lemma II.6.2

A
  1. Prop 6.1: Every function f in bar(L)^0+ induces a measure mü_f on A via mü_f(A):=S f*I_A dmü for all A in A, and we have for any set A in A that mü(A)=0 implies mü_f(A)=0.
    Prop 6.2: Suppose mü is sig-finite and f,g are in bar(L)^0    +. If mü_f=mü_g, then f=g mü-a.e.
    pg. 91
43
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43 Definition II.6.4:
Define absolute continuity and total continuity
Hint: abs looks like eps, delta
Proposition II.6.5: How to abs and tot continuity relate?

A
  1. pg 92
44
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44 Theorem II.6.6: first part up to and including existence of g_1.
State Radon-Nikodým.
Outline what we do in first part.

A

44 pg. 93

45
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45 Theorem II.6.6 (Radon-Nikodým): second part of the proof starting at the definition of the set script(G)

A

45

46
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46 Proposition II.6.8
Show that (R(A),||.||v) is a Banach space.
A
  1. pg. 97
47
Q

47 Theorem II.6.11 State the Lebesgue decomposition theorem.

A
  1. pg. 98
48
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48 Proposition II.7.2 State the prop on the linear funcitonal F_g.

A

48 pg. 101

49
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49 Theorem II.7.3 In What sense are L^p and L^q dual spaces if p and q are conjugate?

A

49 pg. 102

50
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50 Proposition III.1.5 Let E_i be a generator of the sig-field Ai on Omega_i. State a further assumption s.t. E=E_1x…xE_n is a generator of A:=A1x…xAn.

A
  1. pg. 109
51
Q

51 Proposition III.1.9 Let Lam!={} be arbitrary and Slam a semifield on Omega_lam, lam in Lam. Then the set S of all finite intersections of the sets X_j_k^-1(B_j_k) for j_k in Lam and B_j_k in S_j_k for k in 1,…,n is a semifield on Omega. In particular, the finite disjoint unions of rectangle cylinders with basis sets in S_lam form a field which generates the product sig-field Prod(sig(S_lam))).

A

51 pg. 111

52
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52 Theorem III.2.6

State the theorem on the existence of the measure mü1xK.

A

52 pg. 116

53
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53 Theorem II.2.8

State the theorem about integrating over mü1xK.

A

53 pg. 117

54
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54 Theorem III.2.11: State Fubini

Proposition III.2.13: Prove that if f^v:Omega–>bar(R) is A^v-measurable then f^V*I_N^c is A-measurable

A
  1. pg 120
55
Q 
55 Define the consistency condition.
Define compact class
Lemma III.3.3 and the required definitions:
Show that family of unions of compact classes form a compact class.
A
  1. pg. 124
56
Q

56 Theorem III.3.5

State the Theorem of Daniell-Kolmogorov

A

56 pg. 126

57
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57 Theorem III.3.7. State the Theorem of Ionescu-Tulcea.

A

57 pg. 129

58
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58 Theorem IV.1.6 State the theorem of Dini

A

58 pg 137

59
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59
Define the F-open sets G(F).
Lemma IV.1.11: Let F be a Stone vector lattice on Omega. For all f in F_+ and all alpha>=0, we then have the following:
1) {f>alpha} is F-open
2) G(F) is n-closed and closed under countable unions
3)sig(G(F))=sig(F):=sig(f; f in F) is the smallest sig-field which makes all the f in F measurable.

A

59 pg. 139

60
Q

60 Theorem IV.1.12: first and second part of the proof
State the theorem of Daniell-Stone
What is done in first and second part of proof?

A

60 pg. 140

61
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61 Theorem IV.1.12: third part of the proof (Daniell-Stone)

What is done in third part of proof?

A

61 pg. 140

62
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62 Theorem IV.1.12: fourth part and fifth part of the proof

What is done in fourth and fifth part of the proof?

A

62 pg. 140

63
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63 Theorem IV.2.6
Define the Baire sig-field.
State the theorem of Riesz.

A

63 pg. 145

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