mathlib documentation

measure_theory.measure.lebesgue

Lebesgue measure on the real line and on `ℝⁿ` #

We construct Lebesgue measure on the real line, as a particular case of Stieltjes measure associated to the function x ↦ x. We obtain as a consequence Lebesgue measure on ℝⁿ. We prove that they are translation invariant.

We show that, on ℝⁿ, a linear map acts on Lebesgue measure by rescaling it through the absolute value of its determinant, in real.map_linear_map_volume_pi_eq_smul_volume_pi.

More properties of the Lebesgue measure are deduced from this in haar_lebesgue.lean, where they are proved more generally for any additive Haar measure on a finite-dimensional real vector space.

Definition of the Lebesgue measure and lengths of intervals #

@[instance]

def real.measure_space :

measure_theory.measure_space ℝ

Lebesgue measure on the Borel sigma algebra, giving measure b - a to the interval [a, b].

Equations

real.measure_space = {to_measurable_space := real.measurable_space, volume := stieltjes_function.id.measure}

theorem real.volume_val (s : set ℝ) :

⇑measure_theory.measure_space.volume s = ⇑(stieltjes_function.id.measure) s

@[simp]

theorem real.volume_Ico {a b : ℝ} :

⇑measure_theory.measure_space.volume (set.Ico a b) = ennreal.of_real (b - a)

@[simp]

theorem real.volume_Icc {a b : ℝ} :

⇑measure_theory.measure_space.volume (set.Icc a b) = ennreal.of_real (b - a)

@[simp]

theorem real.volume_Ioo {a b : ℝ} :

⇑measure_theory.measure_space.volume (set.Ioo a b) = ennreal.of_real (b - a)

@[simp]

theorem real.volume_Ioc {a b : ℝ} :

⇑measure_theory.measure_space.volume (set.Ioc a b) = ennreal.of_real (b - a)

@[simp]

theorem real.volume_singleton {a : ℝ} :

⇑measure_theory.measure_space.volume {a} = 0

@[simp]

theorem real.volume_univ :

⇑measure_theory.measure_space.volume set.univ = ⊤

@[simp]

theorem real.volume_ball (a r : ℝ) :

⇑measure_theory.measure_space.volume (metric.ball a r) = ennreal.of_real (2 * r)

@[simp]

theorem real.volume_closed_ball (a r : ℝ) :

⇑measure_theory.measure_space.volume (metric.closed_ball a r) = ennreal.of_real (2 * r)

@[simp]

theorem real.volume_emetric_ball (a : ℝ) (r : ℝ≥0∞) :

⇑measure_theory.measure_space.volume (emetric.ball a r) = 2 * r

@[simp]

theorem real.volume_emetric_closed_ball (a : ℝ) (r : ℝ≥0∞) :

⇑measure_theory.measure_space.volume (emetric.closed_ball a r) = 2 * r

@[instance]

def real.has_no_atoms_volume :

measure_theory.has_no_atoms measure_theory.measure_space.volume

@[simp]

theorem real.volume_interval {a b : ℝ} :

⇑measure_theory.measure_space.volume [a, b] = ennreal.of_real |b - a|

@[simp]

theorem real.volume_Ioi {a : ℝ} :

⇑measure_theory.measure_space.volume (set.Ioi a) = ⊤

@[simp]

theorem real.volume_Ici {a : ℝ} :

⇑measure_theory.measure_space.volume (set.Ici a) = ⊤

@[simp]

theorem real.volume_Iio {a : ℝ} :

⇑measure_theory.measure_space.volume (set.Iio a) = ⊤

@[simp]

theorem real.volume_Iic {a : ℝ} :

⇑measure_theory.measure_space.volume (set.Iic a) = ⊤

@[instance]

def real.locally_finite_volume :

measure_theory.is_locally_finite_measure measure_theory.measure_space.volume

@[instance]

def real.is_finite_measure_restrict_Icc (x y : ℝ) :

measure_theory.is_finite_measure (measure_theory.measure_space.volume.restrict (set.Icc x y))

@[instance]

def real.is_finite_measure_restrict_Ico (x y : ℝ) :

measure_theory.is_finite_measure (measure_theory.measure_space.volume.restrict (set.Ico x y))

@[instance]

def real.is_finite_measure_restrict_Ioc (x y : ℝ) :

measure_theory.is_finite_measure (measure_theory.measure_space.volume.restrict (set.Ioc x y))

@[instance]

def real.is_finite_measure_restrict_Ioo (x y : ℝ) :

measure_theory.is_finite_measure (measure_theory.measure_space.volume.restrict (set.Ioo x y))

Volume of a box in `ℝⁿ` #

theorem real.volume_Icc_pi {ι : Type u_1} [fintype ι] {a b : ι → ℝ} :

⇑measure_theory.measure_space.volume (set.Icc a b) = ∏ (i : ι), ennreal.of_real (b i - a i)

@[simp]

theorem real.volume_Icc_pi_to_real {ι : Type u_1} [fintype ι] {a b : ι → ℝ} (h : a ≤ b) :

(⇑measure_theory.measure_space.volume (set.Icc a b)).to_real = ∏ (i : ι), (b i - a i)

theorem real.volume_pi_Ioo {ι : Type u_1} [fintype ι] {a b : ι → ℝ} :

⇑measure_theory.measure_space.volume (set.univ.pi (λ (i : ι), set.Ioo (a i) (b i))) = ∏ (i : ι), ennreal.of_real (b i - a i)

@[simp]

theorem real.volume_pi_Ioo_to_real {ι : Type u_1} [fintype ι] {a b : ι → ℝ} (h : a ≤ b) :

(⇑measure_theory.measure_space.volume (set.univ.pi (λ (i : ι), set.Ioo (a i) (b i)))).to_real = ∏ (i : ι), (b i - a i)

theorem real.volume_pi_Ioc {ι : Type u_1} [fintype ι] {a b : ι → ℝ} :

⇑measure_theory.measure_space.volume (set.univ.pi (λ (i : ι), set.Ioc (a i) (b i))) = ∏ (i : ι), ennreal.of_real (b i - a i)

@[simp]

theorem real.volume_pi_Ioc_to_real {ι : Type u_1} [fintype ι] {a b : ι → ℝ} (h : a ≤ b) :

(⇑measure_theory.measure_space.volume (set.univ.pi (λ (i : ι), set.Ioc (a i) (b i)))).to_real = ∏ (i : ι), (b i - a i)

theorem real.volume_pi_Ico {ι : Type u_1} [fintype ι] {a b : ι → ℝ} :

⇑measure_theory.measure_space.volume (set.univ.pi (λ (i : ι), set.Ico (a i) (b i))) = ∏ (i : ι), ennreal.of_real (b i - a i)

@[simp]

theorem real.volume_pi_Ico_to_real {ι : Type u_1} [fintype ι] {a b : ι → ℝ} (h : a ≤ b) :

(⇑measure_theory.measure_space.volume (set.univ.pi (λ (i : ι), set.Ico (a i) (b i)))).to_real = ∏ (i : ι), (b i - a i)

@[simp]

theorem real.volume_pi_ball {ι : Type u_1} [fintype ι] (a : ι → ℝ) {r : ℝ} (hr : 0 < r) :

⇑measure_theory.measure_space.volume (metric.ball a r) = ennreal.of_real ((2 * r) ^ fintype.card ι)

@[simp]

theorem real.volume_pi_closed_ball {ι : Type u_1} [fintype ι] (a : ι → ℝ) {r : ℝ} (hr : 0 ≤ r) :

⇑measure_theory.measure_space.volume (metric.closed_ball a r) = ennreal.of_real ((2 * r) ^ fintype.card ι)

theorem real.volume_le_diam (s : set ℝ) :

⇑measure_theory.measure_space.volume s ≤ emetric.diam s

theorem real.volume_pi_le_prod_diam {ι : Type u_1} [fintype ι] (s : set (ι → ℝ)) :

⇑measure_theory.measure_space.volume s ≤ ∏ (i : ι), emetric.diam (function.eval i '' s)

theorem real.volume_pi_le_diam_pow {ι : Type u_1} [fintype ι] (s : set (ι → ℝ)) :

⇑measure_theory.measure_space.volume s ≤ emetric.diam s ^ fintype.card ι

Images of the Lebesgue measure under translation/multiplication in ℝ #

theorem real.map_volume_add_left (a : ℝ) :

⇑(measure_theory.measure.map (has_add.add a)) measure_theory.measure_space.volume = measure_theory.measure_space.volume

@[simp]

theorem real.volume_preimage_add_left (a : ℝ) (s : set ℝ) :

⇑measure_theory.measure_space.volume (has_add.add a ⁻¹' s) = ⇑measure_theory.measure_space.volume s

theorem real.map_volume_add_right (a : ℝ) :

⇑(measure_theory.measure.map (λ (_x : ℝ), _x + a)) measure_theory.measure_space.volume = measure_theory.measure_space.volume

@[simp]

theorem real.volume_preimage_add_right (a : ℝ) (s : set ℝ) :

⇑measure_theory.measure_space.volume ((λ (_x : ℝ), _x + a) ⁻¹' s) = ⇑measure_theory.measure_space.volume s

theorem real.smul_map_volume_mul_left {a : ℝ} (h : a ≠ 0) :

ennreal.of_real |a| • ⇑(measure_theory.measure.map (has_mul.mul a)) measure_theory.measure_space.volume = measure_theory.measure_space.volume

theorem real.map_volume_mul_left {a : ℝ} (h : a ≠ 0) :

⇑(measure_theory.measure.map (has_mul.mul a)) measure_theory.measure_space.volume = ennreal.of_real |a⁻¹ | • measure_theory.measure_space.volume

@[simp]

theorem real.volume_preimage_mul_left {a : ℝ} (h : a ≠ 0) (s : set ℝ) :

⇑measure_theory.measure_space.volume (has_mul.mul a ⁻¹' s) = (ennreal.of_real |a⁻¹ |) * ⇑measure_theory.measure_space.volume s

theorem real.smul_map_volume_mul_right {a : ℝ} (h : a ≠ 0) :

ennreal.of_real |a| • ⇑(measure_theory.measure.map (λ (_x : ℝ), _x * a)) measure_theory.measure_space.volume = measure_theory.measure_space.volume

theorem real.map_volume_mul_right {a : ℝ} (h : a ≠ 0) :

⇑(measure_theory.measure.map (λ (_x : ℝ), _x * a)) measure_theory.measure_space.volume = ennreal.of_real |a⁻¹ | • measure_theory.measure_space.volume

@[simp]

theorem real.volume_preimage_mul_right {a : ℝ} (h : a ≠ 0) (s : set ℝ) :

⇑measure_theory.measure_space.volume ((λ (_x : ℝ), _x * a) ⁻¹' s) = (ennreal.of_real |a⁻¹ |) * ⇑measure_theory.measure_space.volume s

@[simp]

theorem real.map_volume_neg :

⇑(measure_theory.measure.map has_neg.neg) measure_theory.measure_space.volume = measure_theory.measure_space.volume

Images of the Lebesgue measure under translation/linear maps in ℝⁿ #

theorem real.map_volume_pi_add_left {ι : Type u_1} [fintype ι] (a : ι → ℝ) :

⇑(measure_theory.measure.map (has_add.add a)) measure_theory.measure_space.volume = measure_theory.measure_space.volume

@[simp]

theorem real.volume_pi_preimage_add_left {ι : Type u_1} [fintype ι] (a : ι → ℝ) (s : set (ι → ℝ)) :

⇑measure_theory.measure_space.volume (has_add.add a ⁻¹' s) = ⇑measure_theory.measure_space.volume s

theorem real.smul_map_diagonal_volume_pi {ι : Type u_1} [fintype ι] [decidable_eq ι] {D : ι → ℝ} (h : (matrix.diagonal D).det ≠ 0) :

ennreal.of_real |(matrix.diagonal D).det| • ⇑(measure_theory.measure.map ⇑(⇑matrix.to_lin' (matrix.diagonal D))) measure_theory.measure_space.volume = measure_theory.measure_space.volume

A diagonal matrix rescales Lebesgue according to its determinant. This is a special case of real.map_matrix_volume_pi_eq_smul_volume_pi, that one should use instead (and whose proof uses this particular case).

theorem real.map_transvection_volume_pi {ι : Type u_1} [fintype ι] [decidable_eq ι] (t : matrix.transvection_struct ι ℝ) :

⇑(measure_theory.measure.map ⇑(⇑matrix.to_lin' t.to_matrix)) measure_theory.measure_space.volume = measure_theory.measure_space.volume

A transvection preserves Lebesgue measure.

theorem real.map_matrix_volume_pi_eq_smul_volume_pi {ι : Type u_1} [fintype ι] [decidable_eq ι] {M : matrix ι ι ℝ} (hM : M.det ≠ 0) :

⇑(measure_theory.measure.map ⇑(⇑matrix.to_lin' M)) measure_theory.measure_space.volume = ennreal.of_real |(M.det)⁻¹ | • measure_theory.measure_space.volume

Any invertible matrix rescales Lebesgue measure through the absolute value of its determinant.

theorem real.map_linear_map_volume_pi_eq_smul_volume_pi {ι : Type u_1} [fintype ι] {f : (ι → ℝ) →ₗ[ℝ ] ι → ℝ} (hf : ⇑linear_map.det f ≠ 0) :

⇑(measure_theory.measure.map ⇑f) measure_theory.measure_space.volume = ennreal.of_real |(⇑linear_map.det f)⁻¹| • measure_theory.measure_space.volume

Any invertible linear map rescales Lebesgue measure through the absolute value of its determinant.

theorem filter.eventually.volume_pos_of_nhds_real {p : ℝ → Prop} {a : ℝ} (h : ∀ᶠ (x : ℝ) in 𝓝 a, p x) :

0 < ⇑measure_theory.measure_space.volume {x : ℝ | p x}

def region_between {α : Type u_1} (f g : α → ℝ) (s : set α) :

set (α × ℝ)

The region between two real-valued functions on an arbitrary set.

Equations

region_between f g s = {p : α × ℝ | p.fst ∈ s ∧ p.snd ∈ set.Ioo (f p.fst) (g p.fst)}

theorem region_between_subset {α : Type u_1} (f g : α → ℝ) (s : set α) :

region_between f g s ⊆ s.prod set.univ

theorem measurable_set_region_between {α : Type u_1} [measurable_space α] {f g : α → ℝ} {s : set α} (hf : measurable f) (hg : measurable g) (hs : measurable_set s) :

measurable_set (region_between f g s)

The region between two measurable functions on a measurable set is measurable.

theorem volume_region_between_eq_lintegral' {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → ℝ} {s : set α} (hf : measurable f) (hg : measurable g) (hs : measurable_set s) :

⇑(μ.prod measure_theory.measure_space.volume) (region_between f g s) = ∫⁻ (y : α) in s, ennreal.of_real ((g - f) y) ∂μ

theorem volume_region_between_eq_lintegral {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → ℝ} {s : set α} [measure_theory.sigma_finite μ] (hf : ae_measurable f (μ.restrict s)) (hg : ae_measurable g (μ.restrict s)) (hs : measurable_set s) :

⇑(μ.prod measure_theory.measure_space.volume) (region_between f g s) = ∫⁻ (y : α) in s, ennreal.of_real ((g - f) y) ∂μ

The volume of the region between two almost everywhere measurable functions on a measurable set can be represented as a Lebesgue integral.

theorem volume_region_between_eq_integral' {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → ℝ} {s : set α} [measure_theory.sigma_finite μ] (f_int : measure_theory.integrable_on f s μ) (g_int : measure_theory.integrable_on g s μ) (hs : measurable_set s) (hfg : f ≤ᵐ[μ.restrict s] g) :

⇑(μ.prod measure_theory.measure_space.volume) (region_between f g s) = ennreal.of_real (∫ (y : α) in s, (g - f) y ∂μ)

theorem volume_region_between_eq_integral {α : Type u_1} [measurable_space α] {μ : measure_theory.measure α} {f g : α → ℝ} {s : set α} [measure_theory.sigma_finite μ] (f_int : measure_theory.integrable_on f s μ) (g_int : measure_theory.integrable_on g s μ) (hs : measurable_set s) (hfg : ∀ (x : α), x ∈ s → f x ≤ g x) :

⇑(μ.prod measure_theory.measure_space.volume) (region_between f g s) = ennreal.of_real (∫ (y : α) in s, (g - f) y ∂μ)

If two functions are integrable on a measurable set, and one function is less than or equal to the other on that set, then the volume of the region between the two functions can be represented as an integral.