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measure_theory.measure.hausdorff

Hausdorff measure and metric (outer) measures #

In this file we define the d-dimensional Hausdorff measure on an (extended) metric space X and the Hausdorff dimension of a set in an (extended) metric space. Let μ d δ be the maximal outer measure such that μ d δ s ≤ (emetric.diam s) ^ d for every set of diameter less than δ. Then the Hausdorff measure μH[d] s of s is defined as ⨆ δ > 0, μ d δ s. By Caratheodory theorem measure_theory.outer_measure.is_metric.borel_le_caratheodory, this is a Borel measure on X.

The value of μH[d], d > 0, on a set s (measurable or not) is given by

μH[d] s =  (r : 0) (hr : 0 < r),  (t :   set X) (hts : s   n, t n)
    (ht :  n, emetric.diam (t n)  r), ∑' n, emetric.diam (t n) ^ d

For every set s for any d < d' we have either μH[d] s = ∞ or μH[d'] s = 0, see measure_theory.measure.hausdorff_measure_zero_or_top. In topology.metric_space.hausdorff_dimension we use this fact to define the Hausdorff dimension dimH of a set in an (extended) metric space.

We also define two generalizations of the Hausdorff measure. In one generalization (see measure_theory.measure.mk_metric) we take any function m (diam s) instead of (diam s) ^ d. In an even more general definition (see measure_theory.measure.mk_metric') we use any function of m : set X → ℝ≥0∞. Some authors start with a partial function m defined only on some sets s : set X (e.g., only on balls or only on measurable sets). This is equivalent to our definition applied to measure_theory.extend m.

We also define a predicate measure_theory.outer_measure.is_metric which says that an outer measure is additive on metric separated pairs of sets: μ (s ∪ t) = μ s + μ t provided that ⨅ (x ∈ s) (y ∈ t), edist x y ≠ 0. This is the property required for the Caratheodory theorem measure_theory.outer_measure.is_metric.borel_le_caratheodory, so we prove this theorem for any metric outer measure, then prove that outer measures constructed using mk_metric' are metric outer measures.

Main definitions #

Main statements #

Basic properties #

Hausdorff measure in ℝⁿ #

Notations #

We use the following notation localized in measure_theory.

Implementation notes #

There are a few similar constructions called the d-dimensional Hausdorff measure. E.g., some sources only allow coverings by balls and use r ^ d instead of (diam s) ^ d. While these construction lead to different Hausdorff measures, they lead to the same notion of the Hausdorff dimension.

Some sources define the 0-dimensional Hausdorff measure to be the counting measure. We define it to be zero on subsingletons because this way we can have a measure.has_no_atoms (measure.hausdorff_measure d) instance.

TODO #

References #

Tags #

Hausdorff measure, measure, metric measure

Metric outer measures #

In this section we define metric outer measures and prove Caratheodory theorem: a metric outer measure has the Caratheodory property.

We say that an outer measure μ in an (e)metric space is metric if μ (s ∪ t) = μ s + μ t for any two metric separated sets s, t.

Equations
theorem measure_theory.outer_measure.is_metric.finset_Union_of_pairwise_separated {ι : Type u_1} {X : Type u_2} [emetric_space X] {μ : measure_theory.outer_measure X} (hm : μ.is_metric) {I : finset ι} {s : ι → set X} (hI : ∀ (i : ι), i I∀ (j : ι), j Ii jis_metric_separated (s i) (s j)) :
μ (⋃ (i : ι) (H : i I), s i) = ∑ (i : ι) in I, μ (s i)

A metric outer measure is additive on a finite set of pairwise metric separated sets.

Caratheodory theorem. If m is a metric outer measure, then every Borel measurable set t is Caratheodory measurable: for any (not necessarily measurable) set s we have μ (s ∩ t) + μ (s \ t) = μ s.

Constructors of metric outer measures #

In this section we provide constructors measure_theory.outer_measure.mk_metric' and measure_theory.outer_measure.mk_metric and prove that these outer measures are metric outer measures. We also prove basic lemmas about map/comap of these measures.

Auxiliary definition for outer_measure.mk_metric': given a function on sets m : set X → ℝ≥0∞, returns the maximal outer measure μ such that μ s ≤ m s for any set s of diameter at most r.

Equations

Given a function m : set X → ℝ≥0∞, mk_metric' m is the supremum of mk_metric'.pre m r over r > 0. Equivalently, it is the limit of mk_metric'.pre m r as r tends to zero from the right.

Equations

Given a function m : ℝ≥0∞ → ℝ≥0∞ and r > 0, let μ r be the maximal outer measure such that μ s = 0 on subsingletons and μ s ≤ m (emetric.diam s) whenever emetric.diam s < r. Then mk_metric m = ⨆ r > 0, μ r. We add ⨆ (hs : ¬s.subsingleton) to ensure that in the case m x = x ^ d the definition gives the expected result for d = 0.

Equations

measure_theory.outer_measure.mk_metric'.pre m r is a trimmed measure provided that m (closure s) = m s for any set s.

An outer measure constructed using outer_measure.mk_metric' is a metric outer measure.

If c ∉ {0, ∞} and m₁ d ≤ c * m₂ d for 0 < d < ε for some ε > 0 (we use ≤ᶠ[𝓝[Ioi 0]] to state this), then mk_metric m₁ hm₁ ≤ c • mk_metric m₂ hm₂.

If m₁ d ≤ m₂ d for 0 < d < ε for some ε > 0 (we use ≤ᶠ[𝓝[Ioi 0]] to state this), then mk_metric m₁ hm₁ ≤ mk_metric m₂ hm₂

theorem measure_theory.outer_measure.le_mk_metric {X : Type u_2} [emetric_space X] (m : ℝ≥0∞ℝ≥0∞) (μ : measure_theory.outer_measure X) (hμ : ∀ (x : X), μ {x} = 0) (r : ℝ≥0∞) (h0 : 0 < r) (hr : ∀ (s : set X), emetric.diam s r¬s.subsingletonμ s m (emetric.diam s)) :

Metric measures #

In this section we use measure_theory.outer_measure.to_measure and theorems about measure_theory.outer_measure.mk_metric'/measure_theory.outer_measure.mk_metric to define measure_theory.measure.mk_metric'/measure_theory.measure.mk_metric. We also restate some lemmas about metric outer measures for metric measures.

Given a function m : set X → ℝ≥0∞, mk_metric' m is the supremum of μ r over r > 0, where μ r is the maximal outer measure μ such that μ s ≤ m s for all s. While each μ r is an outer measure, the supremum is a measure.

Equations

Given a function m : ℝ≥0∞ → ℝ≥0∞, mk_metric m is the supremum of μ r over r > 0, where μ r is the maximal outer measure μ such that μ s ≤ m s for all sets s that contain at least two points. While each mk_metric'.pre is an outer measure, the supremum is a measure.

Equations

If c ∉ {0, ∞} and m₁ d ≤ c * m₂ d for 0 < d < ε for some ε > 0 (we use ≤ᶠ[𝓝[Ioi 0]] to state this), then mk_metric m₁ hm₁ ≤ c • mk_metric m₂ hm₂.

If m₁ d ≤ m₂ d for 0 < d < ε for some ε > 0 (we use ≤ᶠ[𝓝[Ioi 0]] to state this), then mk_metric m₁ hm₁ ≤ mk_metric m₂ hm₂

theorem measure_theory.measure.mk_metric_apply {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] (m : ℝ≥0∞ℝ≥0∞) (s : set X) :
(measure_theory.measure.mk_metric m) s = ⨆ (r : ℝ≥0∞) (hr : 0 < r), ⨅ (t : set X) (hts : s ⋃ (n : ), t n) (ht : ∀ (n : ), emetric.diam (t n) r), ∑' (n : ), ⨆ (ht : ¬(t n).subsingleton), m (emetric.diam (t n))

A formula for measure_theory.measure.mk_metric.

theorem measure_theory.measure.mk_metric_le_liminf_tsum {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {β : Type u_1} {ι : β → Type u_3} [Π (n : β), encodable (ι n)] (s : set X) {l : filter β} (r : β → ℝ≥0∞) (hr : filter.tendsto r l (𝓝 0)) (t : Π (n : β), ι nset X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), emetric.diam (t n i) r n) (hst : ∀ᶠ (n : β) in l, s ⋃ (i : ι n), t n i) (m : ℝ≥0∞ℝ≥0∞) :
(measure_theory.measure.mk_metric m) s l.liminf (λ (n : β), ∑' (i : ι n), m (emetric.diam (t n i)))

To bound the Hausdorff measure (or, more generally, for a measure defined using measure_theory.measure.mk_metric) of a set, one may use coverings with maximum diameter tending to 0, indexed by any sequence of encodable types.

theorem measure_theory.measure.mk_metric_le_liminf_sum {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {β : Type u_1} {ι : β → Type u_3} [hι : Π (n : β), fintype (ι n)] (s : set X) {l : filter β} (r : β → ℝ≥0∞) (hr : filter.tendsto r l (𝓝 0)) (t : Π (n : β), ι nset X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), emetric.diam (t n i) r n) (hst : ∀ᶠ (n : β) in l, s ⋃ (i : ι n), t n i) (m : ℝ≥0∞ℝ≥0∞) :
(measure_theory.measure.mk_metric m) s l.liminf (λ (n : β), ∑ (i : ι n), m (emetric.diam (t n i)))

To bound the Hausdorff measure (or, more generally, for a measure defined using measure_theory.measure.mk_metric) of a set, one may use coverings with maximum diameter tending to 0, indexed by any sequence of finite types.

Hausdorff measure and Hausdorff dimension #

Hausdorff measure on an (e)metric space.

Equations
theorem measure_theory.measure.le_hausdorff_measure {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] (d : ) (μ : measure_theory.measure X) [measure_theory.has_no_atoms μ] (ε : ℝ≥0∞) (h₀ : 0 < ε) (h : ∀ (s : set X), emetric.diam s ε¬s.subsingletonμ s emetric.diam s ^ d) :
theorem measure_theory.measure.hausdorff_measure_apply' {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] (d : ) (s : set X) :
μH[d] s = ⨆ (r : ℝ≥0∞) (hr : 0 < r), ⨅ (t : set X) (hts : s ⋃ (n : ), t n) (ht : ∀ (n : ), emetric.diam (t n) r), ∑' (n : ), ⨆ (ht : ¬(t n).subsingleton), emetric.diam (t n) ^ d

A formula for μH[d] s that works for all d. In case of a positive d a simpler formula is available as measure_theory.measure.hausdorff_measure_apply.

theorem measure_theory.measure.hausdorff_measure_apply {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {d : } (hd : 0 < d) (s : set X) :
μH[d] s = ⨆ (r : ℝ≥0∞) (hr : 0 < r), ⨅ (t : set X) (hts : s ⋃ (n : ), t n) (ht : ∀ (n : ), emetric.diam (t n) r), ∑' (n : ), emetric.diam (t n) ^ d

A formula for μH[d] s that works for all positive d.

theorem measure_theory.measure.hausdorff_measure_le_liminf_tsum {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {β : Type u_1} {ι : β → Type u_3} [hι : Π (n : β), encodable (ι n)] (d : ) (s : set X) {l : filter β} (r : β → ℝ≥0∞) (hr : filter.tendsto r l (𝓝 0)) (t : Π (n : β), ι nset X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), emetric.diam (t n i) r n) (hst : ∀ᶠ (n : β) in l, s ⋃ (i : ι n), t n i) :
μH[d] s l.liminf (λ (n : β), ∑' (i : ι n), emetric.diam (t n i) ^ d)

To bound the Hausdorff measure of a set, one may use coverings with maximum diameter tending to 0, indexed by any sequence of encodable types.

theorem measure_theory.measure.hausdorff_measure_le_liminf_sum {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {β : Type u_1} {ι : β → Type u_3} [hι : Π (n : β), fintype (ι n)] (d : ) (s : set X) {l : filter β} (r : β → ℝ≥0∞) (hr : filter.tendsto r l (𝓝 0)) (t : Π (n : β), ι nset X) (ht : ∀ᶠ (n : β) in l, ∀ (i : ι n), emetric.diam (t n i) r n) (hst : ∀ᶠ (n : β) in l, s ⋃ (i : ι n), t n i) :
μH[d] s l.liminf (λ (n : β), ∑ (i : ι n), emetric.diam (t n i) ^ d)

To bound the Hausdorff measure of a set, one may use coverings with maximum diameter tending to 0, indexed by any sequence of finite types.

theorem measure_theory.measure.hausdorff_measure_zero_or_top {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {d₁ d₂ : } (h : d₁ < d₂) (s : set X) :
μH[d₂] s = 0 μH[d₁] s =

If d₁ < d₂, then for any set s we have either μH[d₂] s = 0, or μH[d₁] s = ∞.

theorem measure_theory.measure.hausdorff_measure_mono {X : Type u_2} [emetric_space X] [measurable_space X] [borel_space X] {d₁ d₂ : } (h : d₁ d₂) (s : set X) :
μH[d₂] s μH[d₁] s

Hausdorff measure μH[d] s is monotone in d.

Hausdorff measure and Lebesgue measure #

@[simp]

In the space ι → ℝ, Hausdorff measure coincides exactly with Lebesgue measure.

Hausdorff measure, Hausdorff dimension, and Hölder or Lipschitz continuous maps #

theorem holder_on_with.hausdorff_measure_image_le {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {C r : ℝ≥0} {f : X → Y} {s : set X} (h : holder_on_with C r f s) (hr : 0 < r) {d : } (hd : 0 d) :
μH[d] (f '' s) (C ^ d) * μH[(r) * d] s

If f : X → Y is Hölder continuous on s with a positive exponent r, then μH[d] (f '' s) ≤ C ^ d * μH[r * d] s.

theorem lipschitz_on_with.hausdorff_measure_image_le {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {K : ℝ≥0} {f : X → Y} {s : set X} (h : lipschitz_on_with K f s) {d : } (hd : 0 d) :
μH[d] (f '' s) (K ^ d) * μH[d] s

If f : X → Y is K-Lipschitz on s, then μH[d] (f '' s) ≤ K ^ d * μH[d] s.

theorem lipschitz_with.hausdorff_measure_image_le {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {K : ℝ≥0} {f : X → Y} (h : lipschitz_with K f) {d : } (hd : 0 d) (s : set X) :
μH[d] (f '' s) (K ^ d) * μH[d] s

If f is a K-Lipschitz map, then it increases the Hausdorff d-measures of sets at most by the factor of K ^ d.

Antilipschitz maps do not decrease Hausdorff measures and dimension #

theorem antilipschitz_with.hausdorff_measure_preimage_le {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {f : X → Y} {K : ℝ≥0} {d : } (hf : antilipschitz_with K f) (hd : 0 d) (s : set Y) :
μH[d] (f ⁻¹' s) (K ^ d) * μH[d] s
theorem antilipschitz_with.le_hausdorff_measure_image {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {f : X → Y} {K : ℝ≥0} {d : } (hf : antilipschitz_with K f) (hd : 0 d) (s : set X) :
μH[d] s (K ^ d) * μH[d] (f '' s)

Isometries preserve the Hausdorff measure and Hausdorff dimension #

theorem isometry.hausdorff_measure_image {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {f : X → Y} {d : } (hf : isometry f) (hd : 0 d function.surjective f) (s : set X) :
μH[d] (f '' s) = μH[d] s
theorem isometry.hausdorff_measure_preimage {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] {f : X → Y} {d : } (hf : isometry f) (hd : 0 d function.surjective f) (s : set Y) :
@[simp]
theorem isometric.hausdorff_measure_image {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] (e : X ≃ᵢ Y) (d : ) (s : set X) :
@[simp]
theorem isometric.hausdorff_measure_preimage {X : Type u_2} {Y : Type u_3} [emetric_space X] [emetric_space Y] [measurable_space X] [borel_space X] [measurable_space Y] [borel_space Y] (e : X ≃ᵢ Y) (d : ) (s : set Y) :