Theorems

Question 1

X, Y countable, X x Y?

Answer 1

If X and Y are countable then X x Y is countable.

Question 2

When is
⋃ Xa
a ϵ A
countable?

Answer 2

If A is countable and Xa is countable for all a in A, then
⋃ Xa
a ϵ A
is countable.

Question 3

Compact subsets of metric spaces

Answer 3

Compact subsets of metric spaces are closed.

Question 4

Closed subsets of compact sets

Answer 4

Closed subsets of compact sets are compact

Question 5

When is ⋂ Ka ≄ ∅

a

Answer 5

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

Question 6

Intersection of nested nonempty compact sets

Answer 6

If {Kn}, n=1 to infinity, is a sequence of nonempty compact sets such that Kn contains K(n+1), then
∞
⋂ Kn ≄ ∅
n=1

Question 7

Heine-Borel and Bolzano-Weirestrass Theorems

Answer 7

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).

Question 8

Basic Properties of Measure

Answer 8

(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→∞

Question 9

Lebesgue measure of a real number

Answer 9

Let x ϵ R. Then 𝜆(x) = 0.

Question 10

Lebesgue measure of a countable subset of R

Answer 10

Let A be a countable subset of R. Then 𝜆(A) = 0.

Question 11

𝜇* an outer measure, when is every Borel subset 𝜇*-measurable?

Answer 11

If 𝜇* is a metric outer measure on a metric space X, then every Borel subset of X is 𝜇*-measurable.

Question 12

Carathéodory’s Theorem

Answer 12

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.

Question 13

Properties of the Cantor set

Answer 13

The Cantor set is compact, nonempty, perfect, and uncountable. 𝜆(C) = 0.

Question 14

Perfect sets in Rⁿ

Answer 14

A perfect set in Rⁿ is uncountable.

Question 15

Real intervals contained in the Cantor set

Answer 15

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.

Question 16

N_𝛿([0, 1])

Answer 16

N_𝛿([0, 1]) = ⌈1/𝛿⌉

Question 17

dimʙ( (0, 1) )

Answer 17

dimʙ( (0, 1) ) = 1

Question 18

dimʙ( [a,b] )

Answer 18

Let a, b be real numbers with a < b. Then dimʙ( [a,b] ) = 1.

Question 19

[0,1] ⊂ Rⁿ where n > 2. dimʙ( [0,1] )

Answer 19

dimʙ( [0,1] ) = 1.

Question 20

F ⊂ Rᵐ ⊂ Rⁿ for integers 0 ⩽ m < n. Properties of lower and upper box dimensions of F.

Answer 20

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.

Question 21

Equivalent definitions of N_𝛿(F)

Answer 21

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.

Question 22

Box dimension of the Cantor set

Answer 22

The Cantor set has box dimension log 2 / log 3

Question 23

Monotonicity property of box dimension

Answer 23

Let E ⊂ F ⊂ Rⁿ. We have the monotonicity property:

lower dimʙ E ⩽ lower dimʙ F

upper dimʙ E ⩽ upper dimʙ F

Question 24

U bounded open subset of Rⁿ. What is dimʙ U?

Answer 24

dimʙ U = n

Question 25

Finite stability property of box dimension

Answer 25

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)

Question 26

Box dimension of Lipschitz mapping

Answer 26

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)

Question 27

Bilipschitz invariance of box dimension

Answer 27

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)

Question 28

Closure invariance of box dimension

Answer 28

Let A ⊂ Rⁿ and Aࠡ be the closure of A. Then
lower dimʙ A = lower dimʙ Aࠡ
upper dimʙ A = upper dimʙ Aࠡ

Question 29

What type of measure is ℋˢ?

Answer 29

Let s ⩾ 0. Then ℋˢ is an outer measure on Rⁿ, and a metric outer measure.

Question 30

What is
∞
lim ∑ ℋˢ_𝛿 (Aⱼ)
𝛿→0 j=1

Answer 30

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

Question 31

Borel sets and ℋˢ

Answer 31

The Borel sets are ℋˢ-measurable. Thus the restriction of ℋˢ to the Borel sets is a measure.

Question 32

Relationship between Hausdorff and Lebesgue measure

Answer 32

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.

Question 33

Hausdorff dim of Lipschitz mappings

Answer 33

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)

Question 34

Equivalent definitions of Hausdorff dimension

Answer 34

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.

Question 35

Properties of Hausdorff dimension

Answer 35

(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))

Question 36

Hausdorrf dimension of the Cantor set

Answer 36

dim_H (C) = log 2 / log 3

Question 37

Hausdorff dim compared to box dim

Answer 37

Let F ⊂ Rⁿ. Then

dim_H (F) ≤ lower dimʙ(F) ≤ upper dimʙ(F)

Question 38

Contractions on closed subsets of Rⁿ

Answer 38

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.

Question 39

Hausdorff metric of contraction on nonempty compact subsets

Answer 39

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)

Question 40

Properties of iterated function systems

Answer 40

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.

Question 41

Open set characterization of continuity

Answer 41

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.

Question 42

Continuous image of a compact set

Answer 42

Suppose f is a continuous function from a compact space X into a space Y. Then f(X) is compact.

Question 43

Hausdorff dimension of self-similar sets satisfying OSC

Answer 43

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) < ∞.

Question 44

Hausdorff dim, box dim of Sierpinski gasket

Answer 44

The Hausdorff dimension and box dimension of the Sierpinski gasket are both
log 3 / log 2

Question 45

Intersections of Player A and Player B’s sets

Answer 45

The intersection
       ∞             ∞
       ⋂ Bₙ   =   ⋂ Aₙ
      n=1          n=1
is a single point in Rⁿ

Question 46

When are (𝛼, 𝛽)-winning sets (𝛼’, 𝛽’)-winning?

Answer 46

Let 0 < 𝛼, 𝛽, 𝛼’, 𝛽’ < 1, 𝛼𝛽 = 𝛼’𝛽’, and 𝛼’ ≤ 𝛼. Then every (𝛼, 𝛽)-winning set is also (𝛼’, 𝛽’)-winning.

Question 47

Properties of winning sets

Answer 47

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)

Question 48

Sets of well and badly approximable numbers

Answer 48

Well and badly approximable numbers are complementary sets, i.e.
BA = R \ WA

Question 49

When is a real number badly approximable?

Answer 49

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.

Question 50

Is BA winning?

Answer 50

The set BA is 1/2-winning in R

Question 51

Player A’s moves in an (𝛼, 𝛽, BA)-game where a ball Bₖ with centre bₖ and radius 𝜌ₖ occurs.

Answer 51

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}

Question 52

Player A’s moves in an (𝛼, 𝛽, BA)-game where Player B begins with a closed ball of radius 𝜌.

Answer 52

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.

Question 53

Khinchin Theorem

Answer 53

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=1

Question 54

ℋ^(dim_H (BA)) (BA)

Answer 54

ℋ^(dim_H (BA)) (BA) = ℋ¹(BA) = 0

Question 55

WA wrt 𝜓_c

Answer 55

Let 𝜓_c : N ⟶ (0, ∞), q ⟼ c/q for any c > 0. We have that
∞
WA ⊃ ⋂ WA(𝜓_(1/n))
n=1