Theorems Flashcards
X, Y countable, X x Y?
If X and Y are countable then X x Y is countable.
When is
⋃ Xa
a ϵ A
countable?
If A is countable and Xa is countable for all a in A, then
⋃ Xa
a ϵ A
is countable.
Compact subsets of metric spaces
Compact subsets of metric spaces are closed.
Closed subsets of compact sets
Closed subsets of compact sets are compact
When is ⋂ Ka ≄ ∅
a
If {Ka} is a collection of compact subsets of a metric space X such that the intersection of every finite subcollection of {Ka} is nonempty, then
⋂ Ka ≄ ∅
a
Intersection of nested nonempty compact sets
If {Kn}, n=1 to infinity, is a sequence of nonempty compact sets such that Kn contains K(n+1), then
∞
⋂ Kn ≄ ∅
n=1
Heine-Borel and Bolzano-Weirestrass Theorems
If a set E in Rⁿ has one of the following 3 properties, then it has the other two:
(a) E is closed and bounded
(b) E is compact
(c) Every infinite subset of E has a limit point in E.
(Heine-Borel is the equivalence of a and b, Bolzano-Weierstrass is the equivalence of a and c).
Basic Properties of Measure
(a) Monotonicity: If E, F ϵ ℳ and E ⊂ F, then 𝜇(E) ⩽ 𝜇(F).
(b) Subadditivity: If {Ej}, j=1 to ∞, ⊂ ℳ, then
∞ ∞
𝜇( ⋃ Ej ) ⩽ ∑ 𝜇(Ej)
j=1 j=1
(c) Continuity from below: If {Ej}, j=1 to ∞, ⊂ ℳ and E₁ ⊂ E₂ ⊂ …, then
∞
𝜇( ⋃ Ej ) = lim 𝜇(Ej)
j=1 j→∞
(d) Continuity from above: If {Ej}, j=1 to ∞, ⊂ ℳ and E₁ ⊃ E₂ ⊃ …, and 𝜇(E₁) < ∞, then
∞
𝜇( ⋂ Ej ) = lim 𝜇(Ej)
j=1 j→∞
Lebesgue measure of a real number
Let x ϵ R. Then 𝜆(x) = 0.
Lebesgue measure of a countable subset of R
Let A be a countable subset of R. Then 𝜆(A) = 0.
𝜇* an outer measure, when is every Borel subset 𝜇*-measurable?
If 𝜇* is a metric outer measure on a metric space X, then every Borel subset of X is 𝜇*-measurable.
Carathéodory’s Theorem
If 𝜇* is an outer measure on a set X, then the collection ℳ of 𝜇-measurable sets is a 𝜎-algebra, and the restriction of 𝜇 to ℳ is a complete measure.
Properties of the Cantor set
The Cantor set is compact, nonempty, perfect, and uncountable. 𝜆(C) = 0.
Perfect sets in Rⁿ
A perfect set in Rⁿ is uncountable.
Real intervals contained in the Cantor set
There does not exist any pair (a, b) ϵ R x R such that a < b and (a, b) ϵ C. In other words, the Cantor set does not contain any (nontrivial) interval.
N_𝛿([0, 1])
N_𝛿([0, 1]) = ⌈1/𝛿⌉
dimʙ( (0, 1) )
dimʙ( (0, 1) ) = 1
dimʙ( [a,b] )
Let a, b be real numbers with a < b. Then dimʙ( [a,b] ) = 1.
[0,1] ⊂ Rⁿ where n > 2. dimʙ( [0,1] )
dimʙ( [0,1] ) = 1.
F ⊂ Rᵐ ⊂ Rⁿ for integers 0 ⩽ m < n. Properties of lower and upper box dimensions of F.
The lower box dimension of F computed using subsets only in Rᵐ is the same as the lower box dimension of F computed using subsets in Rⁿ.
The upper box dimension of F computed using subsets only in Rᵐ is the same as the upper box dimension of F computed using subsets in Rⁿ.
Equivalent definitions of N_𝛿(F)
The lower and upper box dimensions of F ⊂ Rⁿ are given by the formula using N_𝛿(F) where this is any of the following:
(i) the smallest number of closed balls of radius 𝛿 that cover F
(ii) the smallest number of n-dimensional cubes of side length 𝛿 that cover F
(iii) the largest number of disjoint balls of radius 𝛿 with centres in F.
Box dimension of the Cantor set
The Cantor set has box dimension log 2 / log 3
Monotonicity property of box dimension
Let E ⊂ F ⊂ Rⁿ. We have the monotonicity property:
lower dimʙ E ⩽ lower dimʙ F
upper dimʙ E ⩽ upper dimʙ F
U bounded open subset of Rⁿ. What is dimʙ U?
dimʙ U = n
Finite stability property of box dimension
Let E, F ⊂ Rⁿ. Assume the box dimensions of E and F exist. Then the box dimension of E ⋃ F exists, and we have the finite stability property:
dimʙ (E ⋃ F) = max(dimʙ E, dimʙ F)
This is true for any finite collection, i.e.
dimʙ (E₁ ⋃ … ⋃ Eⱼ) = max(dimʙ E₁, …, dimʙ Eⱼ)
Box dimension of Lipschitz mapping
Let F ⊂ Rⁿ and let f : F ⟶ Rᵐ be a Lipschitz mapping. Then
lower dimʙ F ⩾ lower dimʙ f(F)
upper dimʙ F ⩾ upper dimʙ f(F)
Bilipschitz invariance of box dimension
Let F ⊂ Rⁿ and f : F ⟶ Rᵐ be a bilipschitz mapping. Then
lower dimʙ F = lower dimʙ f(F)
upper dimʙ F = upper dimʙ f(F)
Closure invariance of box dimension
Let A ⊂ Rⁿ and Aࠡ be the closure of A. Then
lower dimʙ A = lower dimʙ Aࠡ
upper dimʙ A = upper dimʙ Aࠡ
What type of measure is ℋˢ?
Let s ⩾ 0. Then ℋˢ is an outer measure on Rⁿ, and a metric outer measure.
What is
∞
lim ∑ ℋˢ_𝛿 (Aⱼ)
𝛿→0 j=1
For any countable family of subsets {Aⱼ} (j = 1 to ∞) of Rⁿ, we have that
∞ ∞
lim ∑ ℋˢ_𝛿 (Aⱼ) = ∑ ℋˢ(Aⱼ)
𝛿→0 j=1 j=1
Borel sets and ℋˢ
The Borel sets are ℋˢ-measurable. Thus the restriction of ℋˢ to the Borel sets is a measure.
Relationship between Hausdorff and Lebesgue measure
Let n ϵ N. There exists a constant 𝛾ₙ > 0 such that 𝛾ₙℋⁿ is Lebesgue measure on Rⁿ.
Note the constant can be precisely computed and is the volume of a ball of radius 1/2 in Rⁿ.
Hausdorff dim of Lipschitz mappings
Let F ⊂ Rⁿ and f : F ⟶ Rᵐ be a Lipschitz mapping, namely that there exists a c > 0 such that
|f(x) - f(y)| ≤ c|x - y| for any x, y ϵ F.
Then, for each s ≥ 0, we have
ℋˢ( f(F) ) ≤ cˢ ℋˢ(F)
Equivalent definitions of Hausdorff dimension
Equivalent definitions can be obtained if in our usual definition we only consider 𝛿-covers by the sets {Uj} that are all closed balls or all open balls.
Properties of Hausdorff dimension
(i) Monotonicity: Let E ⊂ F ⊂ Rⁿ. Then dim_H (E) ≤ dim_H (F)
(ii) Countable stability: Let {Fⱼ}, j = 1 to ∞ be a countable family of subsets of Rⁿ. Then
∞
dim_H ( ⋃ Fⱼ ) = sup{dim_H (Fⱼ) : 1 ≤ j < ∞}
j=1
(iii) Open sets: Let F be a nonempty open set of Rⁿ. Then dim_H (F) = n
(iv) Bilipschitz invariance: Let F ⊂ Rⁿ and f : F ⟶ Rᵐ be a bilipschitz mapping. Then
dim_H (F) = dim_H (f(F))
Hausdorrf dimension of the Cantor set
dim_H (C) = log 2 / log 3
Hausdorff dim compared to box dim
Let F ⊂ Rⁿ. Then
dim_H (F) ≤ lower dimʙ(F) ≤ upper dimʙ(F)
Contractions on closed subsets of Rⁿ
Let D ⊂ Rⁿ be a closed set and S : D ⟶ D be a contraction. Then there exist nonempty compact sets E ⊂ D such that S(E) ⊂ E.
Hausdorff metric of contraction on nonempty compact subsets
Let S be a contraction on D with constant 0 < c < 1 and A, B nonempty compact subsets of D. Then d_H ( S(A), S(B) ) ≤ c d_H (A, B)
Properties of iterated function systems
Let {S₁, …, Sₘ} be a family of contractions on D ⊂ Rⁿ such that
|Sⱼ (x) - Sⱼ (y)| ≤ cⱼ |x - y| x, y ϵ D
with 0 < cⱼ < 1. Then there exists a unique invariant set F for the family and F is a nonempty compact set.
Moreover, if E is a nonempty compact subset of D and S is defined as the following transformation 𝒮 ⟶ 𝒮 :
m
S(E) := ⋃ Sⱼ (E)
j=1
and Sᵏ is recursively defined as the transformation
S⁰(E) = E
Sᵏ(E) = S(Sᵏ⁻¹(E)) for all integers k ≥ 1
then
∞
F = ⋂ Sᵏ(E)
k=1
for any nonempty compact subset E ⊂ D for which Sⱼ (E) ⊂ E for every j.
Open set characterization of continuity
Let (X, dₓ) and (Y, dᵧ) be metric spaces. A function f : X ⟶ Y is continuous on X if and only if f⁻¹(V) is open in X for every open set V in Y.
Continuous image of a compact set
Suppose f is a continuous function from a compact space X into a space Y. Then f(X) is compact.
Hausdorff dimension of self-similar sets satisfying OSC
Let {S₁, …, Sₘ} be a family of similarities on Rⁿ,
|Sⱼ (x) - Sⱼ (y)| = cⱼ |x - y| for any x, y ϵ Rⁿ
satisfying the open set condition. If F is the invariant set, then dim_H F = dimʙ F = s where s is determined by
m
∑ cⱼˢ = 1
j=1
Moreover, the s-dimensional Hausdorff measure is finite, i.e. 0 < ℋˢ(F) < ∞.
Hausdorff dim, box dim of Sierpinski gasket
The Hausdorff dimension and box dimension of the Sierpinski gasket are both
log 3 / log 2
Intersections of Player A and Player B’s sets
The intersection
∞ ∞
⋂ Bₙ = ⋂ Aₙ
n=1 n=1
is a single point in RⁿWhen are (𝛼, 𝛽)-winning sets (𝛼’, 𝛽’)-winning?
Let 0 < 𝛼, 𝛽, 𝛼’, 𝛽’ < 1, 𝛼𝛽 = 𝛼’𝛽’, and 𝛼’ ≤ 𝛼. Then every (𝛼, 𝛽)-winning set is also (𝛼’, 𝛽’)-winning.
Properties of winning sets
1) A countable intersection of 𝛼-winning sets is 𝛼-winning
2) If a set is 𝛼-winning, then it is also 𝛼’-winning for all 0 < 𝛼’ ≤ 𝛼
3) The bilipschitz image of a winning set is winning
4) An 𝛼-winning set in Rⁿ is thick (and thus has full Hausdorff dimension)
Sets of well and badly approximable numbers
Well and badly approximable numbers are complementary sets, i.e.
BA = R \ WA
When is a real number badly approximable?
A real number x is badly approximable if and only if there exists a c > 0 (depending on x) such that, for all rational numbers p/q in lowest terms,
|x - p/q| > c/q²
holds.
Is BA winning?
The set BA is 1/2-winning in R
Player A’s moves in an (𝛼, 𝛽, BA)-game where a ball Bₖ with centre bₖ and radius 𝜌ₖ occurs.
Let 0 < 𝛼, 𝛽 < 1 be chosen so that
𝛾 := 1 + 𝛼𝛽 - 2𝛼 > 0
Let t ϵ N such that
(𝛼𝛽)ᵗ < 𝛾/2
Suppose in an (𝛼, 𝛽, BA)-game, a ball Bₖ with centre bₖ and radius 𝜌ₖ occurs. Then Player A can make moves in such a way that Bₖ₊ₜ is contained in
{x ϵ R : x > bₖ + 𝜌ₖ𝛾/2}Player A’s moves in an (𝛼, 𝛽, BA)-game where Player B begins with a closed ball of radius 𝜌.
Let 0 < 𝛼, 𝛽 < 1 be chosen so that
𝛾 := 1 + 𝛼𝛽 - 2𝛼 > 0
Suppose Player B begins an (𝛼, 𝛽, BA)-game by choosing a closed ball of radius 𝜌. Set
𝛿 = (𝛾/2) min( 𝜌, 𝛼²𝛽²𝛾/8)
Then Player A can force
∞
x := ⋂ Bₙ
n=1
to satisfy
|x - p/q| > 𝛿/q²
for all integers p and all non-zero integers q. Note that 𝛿 does not depend on p/q.Khinchin Theorem
Let 𝜓 : N ⟶ (0, ∞) be a non-increasing function. Then the set
WA(𝜓) := {x ϵ R : |x - p/q| < 𝜓(q) / q for infinitely many rationals p/q where q > 0}
has Lebesgue measure 0 if
∞
∑ 𝜓(q) < ∞
q=1
and its complement, the set
R\WA(𝜓)
has Lebesgue measure zero if
∞
∑ 𝜓(q) = ∞
q=1ℋ^(dim_H (BA)) (BA)
ℋ^(dim_H (BA)) (BA) = ℋ¹(BA) = 0
WA wrt 𝜓_c
Let 𝜓_c : N ⟶ (0, ∞), q ⟼ c/q for any c > 0. We have that
∞
WA ⊃ ⋂ WA(𝜓_(1/n))
n=1
