Real Variables - Final

Question 1

relation

Answer 1

1

Question 2

countable - defn + remark + lemma + theorem

Answer 2

2

Question 3

partial order, linear order, ex, ex

Answer 3

3

Question 4

well-ordered - defn, theorem, well-order, ex, 3 properties

Answer 4

4

Question 5

initial segment

Answer 5

5

Question 6

Principle of Transfinite Induction

Answer 6

6

Question 7

Theorem (union of initial segments)

Answer 7

7

Question 8

order isomorphic - defn, ex, Theorem (+ proof), Corollary, Proposition

Answer 8

8

Question 9

transfinite recursion

Answer 9

9

Question 10

Well-ordering Theorem

Answer 10

10

Question 11

Axiom of Choice

Answer 11

11

Question 12

cardinality - defn, Theorem, Corollary

Answer 12

12

Question 13

Cantor-Schrider-Bernstein Theorem

Answer 13

13

Question 14

card(P(A))

Answer 14

14

Question 15

cardinal arithmetic - defn, Theorem

Answer 15

15

Question 16

Zorn’s Lemma

Answer 16

16

Question 17

Hausdorff Maximal Principle

Answer 17

17

Question 18

important equivalence (TFAE)

Answer 18

18

Question 19

A^|B| - defn, Theorem

Answer 19

19

Question 20

algebra, sigma-algebra, ex, ex

Answer 20

20

Question 21

finitely additive, premeasure, measure, ex, ex, finite, probability, sigma-finite, semi-finite

Answer 21

21

Question 22

disjointification Lemma

Answer 22

22

Question 23

A(E), M(E), Proposition, B_X, G_delta, F_sigma, Proposition (about B_R)

Answer 23

23

Question 24

elementary family - defn, Proposition

Answer 24

24

Question 25

measurable space

Answer 25

25

Question 26

continuous from below/above, continuous from above at 0, Proposition

Answer 26

26

Question 27

complete measure space

Answer 27

27

Question 28

outer measure - defn, proposition, µ* measurable

Answer 28

28

Question 29

Caratheodory + following proposition

Answer 29

29 / 30

Question 30

J1.1 (elementary family)

Answer 30

31

Question 31

µ_F Theorem, completion

Answer 31

32

Question 32

regularity properties of Lebesgue-Stiltjes measure

Answer 32

33

Question 33

Theorem (E ∈M_µ, TFAE)

Answer 33

34

Question 34

cantor set - defn, ex, ex, Theorem

Answer 34

35

Question 35

Fat Cantor sets

Answer 35

36

Question 36

non-measurable set

Answer 36

37

Question 37

J1.17 (weird set C)

Answer 37

38

Question 38

measurable function - defn, proposition, proposition, proposition, proposition

Answer 38

39

Question 39

simple function

Answer 39

40

Question 40

important approximation theorem - 3 parts

Answer 40

41

Question 41

f=g a.e., Proposition

Answer 41

42

Question 42

integral of simple, proposition (6 properties), integral of arbitrary f

Answer 42

43

Question 43

monotone convergence theorem

Answer 43

44

Question 44

Fatou’s lemma

Answer 44

45

Question 45

dominated convergence theorem, v1

Answer 45

46

Question 46

Dini’s Theorem

Answer 46

47

Question 47

µ-integrable, Proposition, Proposition

Answer 47

48

Question 48

Generalized dominated convergence theorem

Answer 48

49

Question 49

norm on L1(µ), Properties (3), Proposition (3), Theorem

Answer 49

50

Question 50

Theorem: Lebesgue Stieltzes vs Riemann

Answer 50

52

Question 51

oscillation, omega(f,x)(epsilon), omega(f,x), Lemma

Answer 51

51

Question 52

D(g) gives existence of integral

Answer 52

53

Question 53

converges in measure, equivalence, Cauchy in measure, Theorem, Proposition, Theorem

Answer 53

54

Question 54

almost uniformly, Theorem

Answer 54

55

Question 55

Egoroff’s Theorem

Answer 55

56

Question 56

metric on L_C(\M)

Answer 56

57

Question 57

types of convergence (refresher, 6 types), implies

Answer 57

58

Question 58

product of measure spaces, Theorem

Answer 58

59

Question 59

Tonelli’s Theorem

Answer 59

60

Question 60

Fubini’s Theorem

Answer 60

61

Question 61

Approximation properties of m^n

Answer 61

62

Question 62

uniqueness of Haar measure on R^n

Answer 62

63

Question 63

signed measure

Answer 63

64

Question 64

positive/negative/null for nu

Answer 64

65

Question 65

Hahn-Decomposition Theorem

Answer 65

66

Question 66

mutually singular (perp)

Answer 66

67

Question 67

Jordan Decomposition

Answer 67

68

Question 68

absolutely continuous wrt

Answer 68

69

Question 69

Lebesgue Decomposition Theorem

Answer 69

70

Question 70

Radon-Nikodyn Theorem + Theorem (epsilon-delta) + Proposition (2 parts)

Answer 70

71

Question 71

covering lemma

Answer 71

72

Question 72

locally integrable + Ar(f) + Hardy-Littlewood Maximal function + Lemma + Theorem (epsilon) + Theorem (limit)

Answer 72

73

Question 73

Lebesuge set of f + Theorem + Comment 1 + Lebesgue Density Theorem

Answer 73

74

Question 74

shrinks nicely + Lebesgue Differential Theorem

Answer 74

75

Question 75

Theorem (limit r->0)

Answer 75

76

Question 76

differentiation on R (3 parts)

Answer 76

77

Question 77

bounded variation + remarks + 10 properties + NBV + Theorem

Answer 77

78

Question 78

F absolutely continuous + Theorem (NBV) + Propositon (3 parts) + Theorem (integral)

Answer 78

79