Semester 2 Definitions Flashcards
A = #B means
Bijective
A =< #B
Injective
A finite set
A = null set
A = {1,….,N} for some N £ Natural numbers, N > 0
Countable
A =< #Natural numbers
Countably infinite
A = #Natural numbers
Uncountable
A >= #Natural numbers and #A =/ #Natural numbers
Limit superior
The limit superior limsup(Xn) is the largest limit point of Xn
Limit inferior
Is the smallest limit point of Xn
Sigma-Algebra
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.
A Borel Sigma-Algebra
The Borel σ-algebra on Ω is the σ-algebra generated by τ . (B_Ω :=< τ >σ)
Its elements are called Borel sets
Content
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)
Inner Content u-
Consider a content space (&,B,u) and for any subset A £ P(&):
u-(A) := sup{ u(A-) | A- £ B, A- C A }
Outer Content u+
Consider a content space (&,B,u) and for any subset A £ P(&):
u+ := inf{ u(A+) | A+ £ B, A C A+ }
Interval
An interval I ⊆ Rn, n ∈ N is a set of the form I=I1 × … × In with intervals Ij ⊆ R, j ∈ {1, …, n}
Extended Reals
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
Completion
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-
Inner and Outer Jordan Contents
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]
Sum of intervals
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.
Outer Lebesgue measure
For A ⊂ Rn we let
λ(A) := inf{SUM Ij is a sum of intervals, A ⊂ Union Ij}
λ is called outer Lebesgue measure.
Outer measure
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)
u-Nullset
If µ : P(Ω) → R is an outer measure, then A ⊂ Ω is a µ-nullset if µ(A) = 0
u-Almost everywhere
A property is said to hold µ-almost everywhere iff it holds ∀x ∈ B with B ⊂ Ω, µ(B^c) = 0
sigma-additive (content space)
u is called sigma-additive if for any pairwise disjoint An, n £ N,
u(U An) = E u(An)
u-Finite (content space)
u is called Finite if for any A £ B, u(A) < inf
sigma-finite (content space)
σ-finite if Ω = UNION Ωn for some Ωn ∈ B with µ(Ωn) < ∞ ∀n ∈ N
Premeasure (content space)
A premeasure on B is a content which is sigma additive. It is called a measure if B is a sigma-algebra
u-measurable
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)
Lebesgue Measure
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.
Vitali Sets
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/∼}
Partition
Let B c Rn be measurable. A partition # = (Am) of B is given by pairwise disjoint Am £ L such that B = U Am
Refinement
If #, #~ £ Pb, then #~ is a refinement of #, if # = (Am), #~ = (A~m~) for every m~ £ N, there exists m £ N: A~m~ c Am
Common Refinement
. #~ := (Am n A~m~) is a common refinement of #,#~
Upper sum
Let B c Rn with B £ L and # £ Pb, # = (Am).
Let f:B -> Rinf.
S(# ; f) := E L(Am).supf(Am)
Lower sum
Let B c Rn with B £ L and # £ Pb, # = (Am).
Let f:B -> Rinf.
s(# ; f) := E L(Am).inf f(Am)
Intermediate sum
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($m)
Upper Lebesgue Integral
Upper integral bounded by B of f(x) dL :=
sup{s(π; f)|π ≻ π⋆}
Lower Lebesgue Integral
Let B £ L and f:B -> Rinf is integrable on B if condition * holds and our integral exists such that for any e>0 there exists #e £ Pb such that, #e > #* and all # £ Pb with # > #e.
|o(#,$,f) - int f(x) dL|< e.
sup{ s(#,f)|# > #*}
Characteristic function
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}
Measurable
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
Step Function
If B £ L then f: B -> Rinf is a step function iff it only takes countably many values
Simple Function
A function f: Rinf -> R is called simple if it is measurable and it takes only finitely many values
Unsigned simple
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)
Boolean Algebra
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
Measure space
If & is a set, B a sigma-algebra and u a measure then (&,B,u) is a measure space
Content space
Is & is a set, B is a boolean algebra and u a content, then (&,B,u) is a content space
Power Set
The set of all subsets of &
Complement
The complement of A in Ω is A^c
:= {x ∈ Ω|x /∈ A}
Difference
The difference is A\B := {x ∈ A|x /∈ B}
Jordan algebra
Fundamental theorem on the Lebesgue Measure
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.
