Semester 2 Definitions

Question 1

A = #B means

Answer 1

Bijective

Question 2

A =< #B

Answer 2

Injective

Question 3

A finite set

Answer 3

A = null set
A = {1,….,N} for some N £ Natural numbers, N > 0

Question 4

Countable

Answer 4

A =< #Natural numbers

Question 5

Countably infinite

Answer 5

A = #Natural numbers

Question 6

Uncountable

Answer 6

A >= #Natural numbers and #A =/ #Natural numbers

Question 7

Limit superior

Answer 7

The limit superior limsup(Xn) is the largest limit point of Xn

Question 8

Limit inferior

Answer 8

Is the smallest limit point of Xn

Question 9

Sigma-Algebra

Answer 9

A σ-algebra on a set Ω is a B ∈ P(P(Ω)) such that:
(a) ∅ ∈ B
(b) if A ∈ B, then Ω\A is in B
(c) If An is a countable subset of B, then the union of all An is in B.

Question 10

A Borel Sigma-Algebra

Answer 10

The Borel σ-algebra on Ω is the σ-algebra generated by τ . (B_Ω :=< τ >σ)
Its elements are called Borel sets

Question 11

Content

Answer 11

Suppose B is a Boolean algebra on a set Ω. A content on (Ω, B) is a function µ : B → [0, +∞] such that
(a) µ(∅) = 0
(b) If A, B ∈ B and A ∩ B = ∅ then µ(A ∪ B) = µ(A) + µ(B)

Question 12

Inner Content u-

Answer 12

Consider a content space (&,B,u) and for any subset A £ P(&):
u-(A) := sup{ u(A-) | A- £ B, A- C A }

Question 13

Outer Content u+

Answer 13

Consider a content space (&,B,u) and for any subset A £ P(&):
u+ := inf{ u(A+) | A+ £ B, A C A+ }

Question 14

Interval

Answer 14

An interval I ⊆ Rn, n ∈ N is a set of the form I=I1 × … × In with intervals Ij ⊆ R, j ∈ {1, …, n}

Question 15

Extended Reals

Answer 15

The extended reals are obtained from R by adding two distinct points +- inf.
For any x £ R: -inf < x < +inf
For any x £ Rinf: x.0 = 0, x/inf = 0
if x > -inf : x + inf + inf
if x > 0 : x . inf = inf
-(inf) = - inf, -(-inf) = inf

Question 16

Completion

Answer 16

If ($,B,u) is a content space with u-,u+ as the inner and outer content, then (%,B^,u^) is the completion of ($,B,u) with B^ as a Boolean algebra with B c B^ and u^ := u+ = u-

Question 17

Inner and Outer Jordan Contents

Answer 17

Let A c Rn;
L-(A) := sup{ L^n(A-)|A- c A, A- £ I}
L+(A):= inf{ L^n(A+)|A c A+, A+ £ I}
£ [0,inf]

Question 18

Sum of intervals

Answer 18

If J is a countable set, and for every j ∈ J, Ij ⊂Rn is an interval with λ^n(Ij ) < ∞, then UNION Ij is called a sum of intervals.

Question 19

Outer Lebesgue measure

Answer 19

For A ⊂ Rn we let
λ(A) := inf{SUM Ij is a sum of intervals, A ⊂ Union Ij}
λ is called outer Lebesgue measure.

Question 20

Outer measure

Answer 20

An outer measure on Ω is a map µ : P(Ω) → R
such that
*µ(∅) = 0
* monotony: if A ⊂ B ⊂ Ω, then µ(A) ≤ µ(B)
* σ-subadditivity: if An ∈ P(Ω) ∀n ∈ N, then
u(UNION An) =< SUM u(An)

Question 21

u-Nullset

Answer 21

If µ : P(Ω) → R is an outer measure, then A ⊂ Ω is a µ-nullset if µ(A) = 0

Question 22

u-Almost everywhere

Answer 22

A property is said to hold µ-almost everywhere iff it holds ∀x ∈ B with B ⊂ Ω, µ(B^c) = 0

Question 23

sigma-additive (content space)

Answer 23

u is called sigma-additive if for any pairwise disjoint An, n £ N,
u(U An) = E u(An)

Question 24

u-Finite (content space)

Answer 24

u is called Finite if for any A £ B, u(A) < inf

Question 25

sigma-finite (content space)

Answer 25

σ-finite if Ω = UNION Ωn for some Ωn ∈ B with µ(Ωn) < ∞ ∀n ∈ N

Question 26

Premeasure (content space)

Answer 26

A premeasure on B is a content which is sigma additive. It is called a measure if B is a sigma-algebra

Question 27

u-measurable

Answer 27

Let u:P(&) -> Rinf denote an outer measure on the set &.
Then A £ P(&) is called u-measurable if
for any S £ P(&): u(S) >= u(SnA) + (SnA^c)

Question 28

Lebesgue Measure

Answer 28

If µ = λ is the outer Lebesgue measure on Ω=Rn then
L := {A ∈ P(Rn)|A is λ-measurable},
is the set of Lebesgue measurable sets, λ|L is the Lebesgue Measure.

Question 29

Vitali Sets

Answer 29

On Ω := [0, 1], let x ∼ y : iff x − y ∈ Q. For every ξ ∈ Ω/∼, choose a representative x ∈ Ω, ξ =[xξ],
V := {xξ|ξ ∈ Ω/∼}

Question 30

Partition

Answer 30

Let B c Rn be measurable. A partition # = (Am) of B is given by pairwise disjoint Am £ L such that B = U Am

Question 31

Refinement

Answer 31

If #, #~ £ Pb, then #~ is a refinement of #, if # = (Am), #~ = (A~m~) for every m~ £ N, there exists m £ N: A~m~ c Am

Question 32

Common Refinement

Answer 32

. #~ := (Am n A~m~) is a common refinement of #,#~

Question 33

Upper sum

Answer 33

Let B c Rn with B £ L and # £ Pb, # = (Am).
Let f:B -> Rinf.
S(# ; f) := E L(Am).supf(Am)

Question 34

Lower sum

Answer 34

Let B c Rn with B £ L and # £ Pb, # = (Am).
Let f:B -> Rinf.
s(# ; f) := E L(Am).inf f(Am)

Question 35

Intermediate sum

Answer 35

Let B c Rn with B £ L and # £ Pb, # = (Am).
Let f:B -> Rinf and for any m £ N, choose $m £ Am, $ = ($m).
o(#,$;f) = E L(Am).f($m)

Question 36

Upper Lebesgue Integral

Answer 36

Upper integral bounded by B of f(x) dL :=
sup{s(π; f)|π ≻ π⋆}

Question 37

Lower Lebesgue Integral

Answer 37

Let B £ L and f:B -> Rinf is integrable on B if condition * holds and our integral exists such that for any e>0 there exists #e £ Pb such that, #e > #* and all # £ Pb with # > #e.
|o(#,$,f) - int f(x) dL|< e.
sup{ s(#,f)|# > #*}

Question 38

Characteristic function

Answer 38

For any set & and A £ P(&), the characteristic function for A in & is:
Xa: & -> R,
Xa(L): = {1 if x £ A | 0 if x £/ A}

Question 39

Measurable

Answer 39

Let A,B denote sigma-algebras on the sets &,% and let f: & -> %.
Then f is A-B measurable if for any D £ B: f-1(D) £ A

Question 40

Step Function

Answer 40

If B £ L then f: B -> Rinf is a step function iff it only takes countably many values

Question 41

Simple Function

Answer 41

A function f: Rinf -> R is called simple if it is measurable and it takes only finitely many values

Question 42

Unsigned simple

Answer 42

A function f: Rinf -> R is called simple if it is measurable and it takes only finitely many values. It is unsigned simple if f(Rn) c [0,inf)

Question 43

Boolean Algebra

Answer 43

Let Ω denote a set. A Boolean algebra on Ω is a set B ∈ P(P(Ω)) such that
(a) ∅ ∈ B
(b) if A ∈ B, then Ω\A is in B
(c) if A, B ∈ B, then A ∪ B ∈ B

Question 44

Measure space

Answer 44

If & is a set, B a sigma-algebra and u a measure then (&,B,u) is a measure space

Question 45

Content space

Answer 45

Is & is a set, B is a boolean algebra and u a content, then (&,B,u) is a content space

Question 46

Power Set

Answer 46

The set of all subsets of &

Question 47

Complement

Answer 47

The complement of A in Ω is A^c
:= {x ∈ Ω|x /∈ A}

Question 48

Difference

Answer 48

The difference is A\B := {x ∈ A|x /∈ B}

Question 49

Jordan algebra

Question 50

Fundamental theorem on the Lebesgue Measure

Answer 50

The Lebesgue measurable sets L ⊂ P(Rn) form a σ-algebra J and thus the Borel σ-algebra B_Rn . The Lebesgue measure defines a measure on L, which is invariant under Euclidean motions, and which on bounded. A ∈ J agrees with the Jordan content.