# mathlibdocumentation

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]

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

Equations
theorem real.volume_val (s : set ) :
@[simp]
theorem real.volume_Ico {a b : } :
@[simp]
theorem real.volume_Icc {a b : } :
@[simp]
theorem real.volume_Ioo {a b : } :
@[simp]
theorem real.volume_Ioc {a b : } :
@[simp]
theorem real.volume_singleton {a : } :
@[simp]
@[simp]
theorem real.volume_ball (a r : ) :
@[simp]
theorem real.volume_closed_ball (a r : ) :
@[simp]
theorem real.volume_emetric_ball (a : ) (r : ℝ≥0∞) :
@[simp]
@[simp]
theorem real.volume_interval {a b : } :
@[simp]
theorem real.volume_Ioi {a : } :
@[simp]
theorem real.volume_Ici {a : } :
@[simp]
theorem real.volume_Iio {a : } :
@[simp]
theorem real.volume_Iic {a : } :
@[instance]
@[instance]
@[instance]
@[instance]

### Volume of a box in ℝⁿ#

theorem real.volume_Icc_pi {ι : Type u_1} [fintype ι] {a b : ι → } :
= ∏ (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 → } (h : a b) :
= ∏ (i : ι), (b i - a i)
theorem real.volume_pi_Ioo {ι : Type u_1} [fintype ι] {a b : ι → } :
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 → } (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 → } :
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 → } (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 → } :
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 → } (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 : } (hr : 0 < r) :
= ennreal.of_real ((2 * r) ^
@[simp]
theorem real.volume_pi_closed_ball {ι : Type u_1} [fintype ι] (a : ι → ) {r : } (hr : 0 r) :
theorem real.volume_le_diam (s : set ) :
theorem real.volume_pi_le_prod_diam {ι : Type u_1} [fintype ι] (s : set (ι → )) :
∏ (i : ι), emetric.diam '' s)
theorem real.volume_pi_le_diam_pow {ι : Type u_1} [fintype ι] (s : set (ι → )) :

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

@[simp]
theorem real.volume_preimage_add_left (a : ) (s : set ) :
@[simp]
@[simp]
theorem real.volume_preimage_mul_left {a : } (h : a 0) (s : set ) :
theorem real.smul_map_volume_mul_right {a : } (h : a 0) :
theorem real.map_volume_mul_right {a : } (h : a 0) :
@[simp]
theorem real.volume_preimage_mul_right {a : } (h : a 0) (s : set ) :

### 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 → ) :
@[simp]
theorem real.volume_pi_preimage_add_left {ι : Type u_1} [fintype ι] (a : ι → ) (s : set (ι → )) :
theorem real.smul_map_diagonal_volume_pi {ι : Type u_1} [fintype ι] [decidable_eq ι] {D : ι → } (h : .det 0) :

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 : ) :

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 : ι } (hM : M.det 0) :

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 → ) →ₗ[] ι → } (hf : 0) :

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

theorem filter.eventually.volume_pos_of_nhds_real {p : → Prop} {a : } (h : ∀ᶠ (x : ) in 𝓝 a, p x) :
def region_between {α : Type u_1} (f g : α → ) (s : set α) :
set × )

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

Equations
theorem region_between_subset {α : Type u_1} (f g : α → ) (s : set α) :
s
theorem measurable_set_region_between {α : Type u_1} {f g : α → } {s : set α} (hf : measurable f) (hg : measurable g) (hs : measurable_set s) :

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

theorem volume_region_between_eq_lintegral' {α : Type u_1} {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : measurable f) (hg : measurable g) (hs : measurable_set s) :
= ∫⁻ (y : α) in s, ennreal.of_real ((g - f) y) μ
theorem volume_region_between_eq_lintegral {α : Type u_1} {μ : measure_theory.measure α} {f g : α → } {s : set α} (hf : (μ.restrict s)) (hg : (μ.restrict s)) (hs : measurable_set 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} {μ : measure_theory.measure α} {f g : α → } {s : set α} (f_int : μ) (g_int : μ) (hs : measurable_set s) (hfg : f ≤ᵐ[μ.restrict s] g) :
= ennreal.of_real ( (y : α) in s, (g - f) y μ)
theorem volume_region_between_eq_integral {α : Type u_1} {μ : measure_theory.measure α} {f g : α → } {s : set α} (f_int : μ) (g_int : μ) (hs : measurable_set s) (hfg : ∀ (x : α), x sf x g x) :
= 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.