# Integrable functions and L¹ space #

In the first part of this file, the predicate Integrable is defined and basic properties of integrable functions are proved.

Such a predicate is already available under the name Memℒp 1. We give a direct definition which is easier to use, and show that it is equivalent to Memℒp 1

In the second part, we establish an API between Integrable and the space L¹ of equivalence classes of integrable functions, already defined as a special case of L^p spaces for p = 1.

## Notation #

• α →₁[μ] β is the type of L¹ space, where α is a MeasureSpace and β is a NormedAddCommGroup with a SecondCountableTopology. f : α →ₘ β is a "function" in L¹. In comments, [f] is also used to denote an L¹ function.

₁ can be typed as \1.

## Main definitions #

• Let f : α → β be a function, where α is a MeasureSpace and β a NormedAddCommGroup. Then HasFiniteIntegral f means (∫⁻ a, ‖f a‖₊) < ∞.

• If β is moreover a MeasurableSpace then f is called Integrable if f is Measurable and HasFiniteIntegral f holds.

## Implementation notes #

To prove something for an arbitrary integrable function, a useful theorem is Integrable.induction in the file SetIntegral.

## Tags #

integrable, function space, l1

### Some results about the Lebesgue integral involving a normed group #

theorem MeasureTheory.lintegral_nnnorm_eq_lintegral_edist {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) :
∫⁻ (a : α), f a‖₊μ = ∫⁻ (a : α), edist (f a) 0μ
theorem MeasureTheory.lintegral_norm_eq_lintegral_edist {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) :
∫⁻ (a : α), μ = ∫⁻ (a : α), edist (f a) 0μ
theorem MeasureTheory.lintegral_edist_triangle {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} {h : αβ} (hf : ) (hh : ) :
∫⁻ (a : α), edist (f a) (g a)μ ∫⁻ (a : α), edist (f a) (h a)μ + ∫⁻ (a : α), edist (g a) (h a)μ
theorem MeasureTheory.lintegral_nnnorm_zero {α : Type u_1} {β : Type u_2} {m : } {μ : } :
∫⁻ (x : α), 0‖₊μ = 0
theorem MeasureTheory.lintegral_nnnorm_add_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} (hf : ) (g : αγ) :
∫⁻ (a : α), f a‖₊ + g a‖₊μ = ∫⁻ (a : α), f a‖₊μ + ∫⁻ (a : α), g a‖₊μ
theorem MeasureTheory.lintegral_nnnorm_add_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } (f : αβ) {g : αγ} (hg : ) :
∫⁻ (a : α), f a‖₊ + g a‖₊μ = ∫⁻ (a : α), f a‖₊μ + ∫⁻ (a : α), g a‖₊μ
theorem MeasureTheory.lintegral_nnnorm_neg {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} :
∫⁻ (a : α), (-f) a‖₊μ = ∫⁻ (a : α), f a‖₊μ

### The predicate HasFiniteIntegral#

def MeasureTheory.HasFiniteIntegral {α : Type u_1} {β : Type u_2} :
{x : } → (αβ)

HasFiniteIntegral f μ means that the integral ∫⁻ a, ‖f a‖ ∂μ is finite. HasFiniteIntegral f means HasFiniteIntegral f volume.

Equations
Instances For
theorem MeasureTheory.hasFiniteIntegral_def {α : Type u_1} {β : Type u_2} :
∀ {x : } (f : αβ) (μ : ), ∫⁻ (a : α), f a‖₊μ <
theorem MeasureTheory.hasFiniteIntegral_iff_norm {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) :
∫⁻ (a : α), μ <
theorem MeasureTheory.hasFiniteIntegral_iff_edist {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) :
∫⁻ (a : α), edist (f a) 0μ <
theorem MeasureTheory.hasFiniteIntegral_iff_ofReal {α : Type u_1} {m : } {μ : } {f : α} (h : 0 ≤ᵐ[μ] f) :
∫⁻ (a : α), ENNReal.ofReal (f a)μ <
theorem MeasureTheory.hasFiniteIntegral_iff_ofNNReal {α : Type u_1} {m : } {μ : } {f : αNNReal} :
MeasureTheory.HasFiniteIntegral (fun (x : α) => (f x)) μ ∫⁻ (a : α), (f a)μ <
theorem MeasureTheory.HasFiniteIntegral.mono {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (hg : ) (h : ∀ᵐ (a : α) ∂μ, f a g a) :
theorem MeasureTheory.HasFiniteIntegral.mono' {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : α} (hg : ) (h : ∀ᵐ (a : α) ∂μ, f a g a) :
theorem MeasureTheory.HasFiniteIntegral.congr' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (hf : ) (h : ∀ᵐ (a : α) ∂μ, f a = g a) :
theorem MeasureTheory.hasFiniteIntegral_congr' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (h : ∀ᵐ (a : α) ∂μ, f a = g a) :
theorem MeasureTheory.HasFiniteIntegral.congr {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) (h : f =ᵐ[μ] g) :
theorem MeasureTheory.hasFiniteIntegral_congr {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (h : f =ᵐ[μ] g) :
theorem MeasureTheory.hasFiniteIntegral_const_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {c : β} :
MeasureTheory.HasFiniteIntegral (fun (x : α) => c) μ c = 0 μ Set.univ <
theorem MeasureTheory.hasFiniteIntegral_const {α : Type u_1} {β : Type u_2} {m : } {μ : } (c : β) :
MeasureTheory.HasFiniteIntegral (fun (x : α) => c) μ
theorem MeasureTheory.hasFiniteIntegral_of_bounded {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {C : } (hC : ∀ᵐ (a : α) ∂μ, f a C) :
theorem MeasureTheory.HasFiniteIntegral.of_finite {α : Type u_1} {β : Type u_2} {m : } {μ : } [] {f : αβ} :
@[deprecated MeasureTheory.HasFiniteIntegral.of_finite]
theorem MeasureTheory.hasFiniteIntegral_of_fintype {α : Type u_1} {β : Type u_2} {m : } {μ : } [] {f : αβ} :

Alias of MeasureTheory.HasFiniteIntegral.of_finite.

theorem MeasureTheory.HasFiniteIntegral.mono_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (h : ) (hμ : μ ν) :
theorem MeasureTheory.HasFiniteIntegral.add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (hμ : ) (hν : ) :
theorem MeasureTheory.HasFiniteIntegral.left_of_add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (h : ) :
theorem MeasureTheory.HasFiniteIntegral.right_of_add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (h : ) :
@[simp]
theorem MeasureTheory.hasFiniteIntegral_add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} :
theorem MeasureTheory.HasFiniteIntegral.smul_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (h : ) {c : ENNReal} (hc : c ) :
@[simp]
theorem MeasureTheory.hasFiniteIntegral_zero_measure {α : Type u_1} {β : Type u_2} {m : } (f : αβ) :
@[simp]
theorem MeasureTheory.hasFiniteIntegral_zero (α : Type u_1) (β : Type u_2) {m : } (μ : ) :
MeasureTheory.HasFiniteIntegral (fun (x : α) => 0) μ
theorem MeasureTheory.HasFiniteIntegral.neg {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hfi : ) :
@[simp]
theorem MeasureTheory.hasFiniteIntegral_neg_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} :
theorem MeasureTheory.HasFiniteIntegral.norm {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hfi : ) :
MeasureTheory.HasFiniteIntegral (fun (a : α) => f a) μ
theorem MeasureTheory.hasFiniteIntegral_norm_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) :
MeasureTheory.HasFiniteIntegral (fun (a : α) => f a) μ
theorem MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top {α : Type u_1} {m : } {μ : } {f : αENNReal} (hf : ∫⁻ (x : α), f xμ ) :
MeasureTheory.HasFiniteIntegral (fun (x : α) => (f x).toReal) μ
theorem MeasureTheory.isFiniteMeasure_withDensity_ofReal {α : Type u_1} {m : } {μ : } {f : α} (hfi : ) :
MeasureTheory.IsFiniteMeasure (μ.withDensity fun (x : α) => ENNReal.ofReal (f x))
theorem MeasureTheory.all_ae_ofReal_F_le_bound {α : Type u_1} {β : Type u_2} {m : } {μ : } {F : αβ} {bound : α} (h : ∀ (n : ), ∀ᵐ (a : α) ∂μ, F n a bound a) (n : ) :
∀ᵐ (a : α) ∂μ, ENNReal.ofReal F n a ENNReal.ofReal (bound a)
theorem MeasureTheory.all_ae_tendsto_ofReal_norm {α : Type u_1} {β : Type u_2} {m : } {μ : } {F : αβ} {f : αβ} (h : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun (n : ) => F n a) Filter.atTop (nhds (f a))) :
∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun (n : ) => ENNReal.ofReal F n a) Filter.atTop (nhds ())
theorem MeasureTheory.all_ae_ofReal_f_le_bound {α : Type u_1} {β : Type u_2} {m : } {μ : } {F : αβ} {f : αβ} {bound : α} (h_bound : ∀ (n : ), ∀ᵐ (a : α) ∂μ, F n a bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun (n : ) => F n a) Filter.atTop (nhds (f a))) :
∀ᵐ (a : α) ∂μ, ENNReal.ofReal (bound a)
theorem MeasureTheory.hasFiniteIntegral_of_dominated_convergence {α : Type u_1} {β : Type u_2} {m : } {μ : } {F : αβ} {f : αβ} {bound : α} (bound_hasFiniteIntegral : ) (h_bound : ∀ (n : ), ∀ᵐ (a : α) ∂μ, F n a bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun (n : ) => F n a) Filter.atTop (nhds (f a))) :
theorem MeasureTheory.tendsto_lintegral_norm_of_dominated_convergence {α : Type u_1} {β : Type u_2} {m : } {μ : } {F : αβ} {f : αβ} {bound : α} (F_measurable : ∀ (n : ), ) (bound_hasFiniteIntegral : ) (h_bound : ∀ (n : ), ∀ᵐ (a : α) ∂μ, F n a bound a) (h_lim : ∀ᵐ (a : α) ∂μ, Filter.Tendsto (fun (n : ) => F n a) Filter.atTop (nhds (f a))) :
Filter.Tendsto (fun (n : ) => ∫⁻ (a : α), ENNReal.ofReal F n a - f aμ) Filter.atTop (nhds 0)

Lemmas used for defining the positive part of an L¹ function

theorem MeasureTheory.HasFiniteIntegral.max_zero {α : Type u_1} {m : } {μ : } {f : α} (hf : ) :
MeasureTheory.HasFiniteIntegral (fun (a : α) => max (f a) 0) μ
theorem MeasureTheory.HasFiniteIntegral.min_zero {α : Type u_1} {m : } {μ : } {f : α} (hf : ) :
MeasureTheory.HasFiniteIntegral (fun (a : α) => min (f a) 0) μ
theorem MeasureTheory.HasFiniteIntegral.smul {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [] (c : 𝕜) {f : αβ} :
theorem MeasureTheory.hasFiniteIntegral_smul_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [] [] {c : 𝕜} (hc : ) (f : αβ) :
theorem MeasureTheory.HasFiniteIntegral.const_mul {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.HasFiniteIntegral (fun (x : α) => c * f x) μ
theorem MeasureTheory.HasFiniteIntegral.mul_const {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.HasFiniteIntegral (fun (x : α) => f x * c) μ

### The predicate Integrable#

def MeasureTheory.Integrable {β : Type u_2} {α : Type u_5} :
{x : } → (αβ)

Integrable f μ means that f is measurable and that the integral ∫⁻ a, ‖f a‖ ∂μ is finite. Integrable f means Integrable f volume.

Equations
Instances For
theorem MeasureTheory.memℒp_one_iff_integrable {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} :
theorem MeasureTheory.Integrable.aestronglyMeasurable {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
theorem MeasureTheory.Integrable.aemeasurable {α : Type u_1} {β : Type u_2} {m : } {μ : } [] [] {f : αβ} (hf : ) :
theorem MeasureTheory.Integrable.hasFiniteIntegral {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
theorem MeasureTheory.Integrable.mono {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (hg : ) (hf : ) (h : ∀ᵐ (a : α) ∂μ, f a g a) :
theorem MeasureTheory.Integrable.mono' {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : α} (hg : ) (hf : ) (h : ∀ᵐ (a : α) ∂μ, f a g a) :
theorem MeasureTheory.Integrable.congr' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (hf : ) (hg : ) (h : ∀ᵐ (a : α) ∂μ, f a = g a) :
theorem MeasureTheory.integrable_congr' {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (hf : ) (hg : ) (h : ∀ᵐ (a : α) ∂μ, f a = g a) :
theorem MeasureTheory.Integrable.congr {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) (h : f =ᵐ[μ] g) :
theorem MeasureTheory.integrable_congr {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (h : f =ᵐ[μ] g) :
theorem MeasureTheory.integrable_const_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {c : β} :
MeasureTheory.Integrable (fun (x : α) => c) μ c = 0 μ Set.univ <
@[simp]
theorem MeasureTheory.integrable_const {α : Type u_1} {β : Type u_2} {m : } {μ : } (c : β) :
MeasureTheory.Integrable (fun (x : α) => c) μ
@[simp]
theorem MeasureTheory.Integrable.of_finite {α : Type u_1} {β : Type u_2} [] [] (μ : ) (f : αβ) :
MeasureTheory.Integrable (fun (a : α) => f a) μ
@[deprecated MeasureTheory.Integrable.of_finite]
theorem MeasureTheory.integrable_of_fintype {α : Type u_1} {β : Type u_2} [] [] (μ : ) (f : αβ) :
MeasureTheory.Integrable (fun (a : α) => f a) μ

Alias of MeasureTheory.Integrable.of_finite.

theorem MeasureTheory.Memℒp.integrable_norm_rpow {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {p : ENNReal} (hf : ) (hp_ne_zero : p 0) (hp_ne_top : p ) :
MeasureTheory.Integrable (fun (x : α) => f x ^ p.toReal) μ
theorem MeasureTheory.Memℒp.integrable_norm_rpow' {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {p : ENNReal} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => f x ^ p.toReal) μ
theorem MeasureTheory.Integrable.mono_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (h : ) (hμ : μ ν) :
theorem MeasureTheory.Integrable.of_measure_le_smul {α : Type u_1} {β : Type u_2} {m : } {μ : } {μ' : } (c : ENNReal) (hc : c ) (hμ'_le : μ' c μ) {f : αβ} (hf : ) :
theorem MeasureTheory.Integrable.add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (hμ : ) (hν : ) :
theorem MeasureTheory.Integrable.left_of_add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (h : MeasureTheory.Integrable f (μ + ν)) :
theorem MeasureTheory.Integrable.right_of_add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} (h : MeasureTheory.Integrable f (μ + ν)) :
@[simp]
theorem MeasureTheory.integrable_add_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {ν : } {f : αβ} :
@[simp]
theorem MeasureTheory.integrable_zero_measure {α : Type u_1} {β : Type u_2} :
∀ {x : } {f : αβ},
theorem MeasureTheory.integrable_finset_sum_measure {α : Type u_1} {β : Type u_2} {ι : Type u_5} {m : } {f : αβ} {μ : } {s : } :
MeasureTheory.Integrable f (is, μ i) is, MeasureTheory.Integrable f (μ i)
theorem MeasureTheory.Integrable.smul_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (h : ) {c : ENNReal} (hc : c ) :
theorem MeasureTheory.Integrable.smul_measure_nnreal {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (h : ) {c : NNReal} :
theorem MeasureTheory.integrable_smul_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {c : ENNReal} (h₁ : c 0) (h₂ : c ) :
theorem MeasureTheory.integrable_inv_smul_measure {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {c : ENNReal} (h₁ : c 0) (h₂ : c ) :
theorem MeasureTheory.Integrable.to_average {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (h : ) :
MeasureTheory.Integrable f ((μ Set.univ)⁻¹ μ)
theorem MeasureTheory.integrable_average {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} :
MeasureTheory.Integrable f ((μ Set.univ)⁻¹ μ)
theorem MeasureTheory.integrable_map_measure {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] {f : αδ} {g : δβ} (hg : ) (hf : ) :
theorem MeasureTheory.Integrable.comp_aemeasurable {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] {f : αδ} {g : δβ} (hg : ) (hf : ) :
theorem MeasureTheory.Integrable.comp_measurable {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] {f : αδ} {g : δβ} (hg : ) (hf : ) :
theorem MeasurableEmbedding.integrable_map_iff {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] {f : αδ} (hf : ) {g : δβ} :
theorem MeasureTheory.integrable_map_equiv {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] (f : α ≃ᵐ δ) (g : δβ) :
theorem MeasureTheory.MeasurePreserving.integrable_comp {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] {ν : } {g : δβ} {f : αδ} (hf : ) (hg : ) :
theorem MeasureTheory.MeasurePreserving.integrable_comp_emb {α : Type u_1} {β : Type u_2} {δ : Type u_4} {m : } {μ : } [] {f : αδ} {ν : } (h₁ : ) (h₂ : ) {g : δβ} :
theorem MeasureTheory.lintegral_edist_lt_top {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) (hg : ) :
∫⁻ (a : α), edist (f a) (g a)μ <
@[simp]
theorem MeasureTheory.integrable_zero (α : Type u_1) (β : Type u_2) {m : } (μ : ) :
MeasureTheory.Integrable (fun (x : α) => 0) μ
theorem MeasureTheory.Integrable.add' {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) (hg : ) :
theorem MeasureTheory.Integrable.add {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) (hg : ) :
theorem MeasureTheory.integrable_finset_sum' {α : Type u_1} {β : Type u_2} {m : } {μ : } {ι : Type u_5} (s : ) {f : ιαβ} (hf : is, MeasureTheory.Integrable (f i) μ) :
MeasureTheory.Integrable (is, f i) μ
theorem MeasureTheory.integrable_finset_sum {α : Type u_1} {β : Type u_2} {m : } {μ : } {ι : Type u_5} (s : ) {f : ιαβ} (hf : is, MeasureTheory.Integrable (f i) μ) :
MeasureTheory.Integrable (fun (a : α) => is, f i a) μ
theorem MeasureTheory.Integrable.neg {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
@[simp]
theorem MeasureTheory.integrable_neg_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} :
@[simp]
theorem MeasureTheory.integrable_add_iff_integrable_right {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) :
@[simp]
theorem MeasureTheory.integrable_add_iff_integrable_left {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) :
theorem MeasureTheory.integrable_left_of_integrable_add_of_nonneg {α : Type u_1} {m : } {μ : } {f : α} {g : α} (h_meas : ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g) (h_int : MeasureTheory.Integrable (f + g) μ) :
theorem MeasureTheory.integrable_right_of_integrable_add_of_nonneg {α : Type u_1} {m : } {μ : } {f : α} {g : α} (h_meas : ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g) (h_int : MeasureTheory.Integrable (f + g) μ) :
theorem MeasureTheory.integrable_add_iff_of_nonneg {α : Type u_1} {m : } {μ : } {f : α} {g : α} (h_meas : ) (hf : 0 ≤ᵐ[μ] f) (hg : 0 ≤ᵐ[μ] g) :
theorem MeasureTheory.integrable_add_iff_of_nonpos {α : Type u_1} {m : } {μ : } {f : α} {g : α} (h_meas : ) (hf : f ≤ᵐ[μ] 0) (hg : g ≤ᵐ[μ] 0) :
@[simp]
theorem MeasureTheory.integrable_add_const_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {c : β} :
MeasureTheory.Integrable (fun (x : α) => f x + c) μ
@[simp]
theorem MeasureTheory.integrable_const_add_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {c : β} :
MeasureTheory.Integrable (fun (x : α) => c + f x) μ
theorem MeasureTheory.Integrable.sub {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} {g : αβ} (hf : ) (hg : ) :
theorem MeasureTheory.Integrable.norm {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
MeasureTheory.Integrable (fun (a : α) => f a) μ
theorem MeasureTheory.Integrable.inf {α : Type u_1} {m : } {μ : } {β : Type u_5} {f : αβ} {g : αβ} (hf : ) (hg : ) :
theorem MeasureTheory.Integrable.sup {α : Type u_1} {m : } {μ : } {β : Type u_5} {f : αβ} {g : αβ} (hf : ) (hg : ) :
theorem MeasureTheory.Integrable.abs {α : Type u_1} {m : } {μ : } {β : Type u_5} {f : αβ} (hf : ) :
MeasureTheory.Integrable (fun (a : α) => |f a|) μ
theorem MeasureTheory.Integrable.bdd_mul {α : Type u_1} {m : } {μ : } {F : Type u_5} {f : αF} {g : αF} (hint : ) (hm : ) (hfbdd : ∃ (C : ), ∀ (x : α), f x C) :
MeasureTheory.Integrable (fun (x : α) => f x * g x) μ
theorem MeasureTheory.Integrable.essSup_smul {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [] {f : αβ} (hf : ) {g : α𝕜} (g_aestronglyMeasurable : ) (ess_sup_g : essSup (fun (x : α) => g x‖₊) μ ) :
MeasureTheory.Integrable (fun (x : α) => g x f x) μ

Hölder's inequality for integrable functions: the scalar multiplication of an integrable vector-valued function by a scalar function with finite essential supremum is integrable.

theorem MeasureTheory.Integrable.smul_essSup {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] {f : α𝕜} (hf : ) {g : αβ} (g_aestronglyMeasurable : ) (ess_sup_g : essSup (fun (x : α) => g x‖₊) μ ) :
MeasureTheory.Integrable (fun (x : α) => f x g x) μ

Hölder's inequality for integrable functions: the scalar multiplication of an integrable scalar-valued function by a vector-value function with finite essential supremum is integrable.

theorem MeasureTheory.integrable_norm_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
MeasureTheory.Integrable (fun (a : α) => f a) μ
theorem MeasureTheory.integrable_of_norm_sub_le {α : Type u_1} {β : Type u_2} {m : } {μ : } {f₀ : αβ} {f₁ : αβ} {g : α} (hf₁_m : ) (hf₀_i : ) (hg_i : ) (h : ∀ᵐ (a : α) ∂μ, f₀ a - f₁ a g a) :
theorem MeasureTheory.Integrable.prod_mk {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {f : αβ} {g : αγ} (hf : ) (hg : ) :
MeasureTheory.Integrable (fun (x : α) => (f x, g x)) μ
theorem MeasureTheory.Memℒp.integrable {α : Type u_1} {β : Type u_2} {m : } {μ : } {q : ENNReal} (hq1 : 1 q) {f : αβ} (hfq : ) :
theorem MeasureTheory.Integrable.measure_norm_ge_lt_top {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) {ε : } (hε : 0 < ε) :
μ {x : α | ε f x} <

A non-quantitative version of Markov inequality for integrable functions: the measure of points where ‖f x‖ ≥ ε is finite for all positive ε.

theorem MeasureTheory.Integrable.measure_norm_gt_lt_top {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) {ε : } (hε : 0 < ε) :
μ {x : α | ε < f x} <

A non-quantitative version of Markov inequality for integrable functions: the measure of points where ‖f x‖ > ε is finite for all positive ε.

theorem MeasureTheory.Integrable.measure_ge_lt_top {α : Type u_1} {m : } {μ : } {f : α} (hf : ) {ε : } (ε_pos : 0 < ε) :
μ {a : α | ε f a} <

If f is ℝ-valued and integrable, then for any c > 0 the set {x | f x ≥ c} has finite measure.

theorem MeasureTheory.Integrable.measure_le_lt_top {α : Type u_1} {m : } {μ : } {f : α} (hf : ) {c : } (c_neg : c < 0) :
μ {a : α | f a c} <

If f is ℝ-valued and integrable, then for any c < 0 the set {x | f x ≤ c} has finite measure.

theorem MeasureTheory.Integrable.measure_gt_lt_top {α : Type u_1} {m : } {μ : } {f : α} (hf : ) {ε : } (ε_pos : 0 < ε) :
μ {a : α | ε < f a} <

If f is ℝ-valued and integrable, then for any c > 0 the set {x | f x > c} has finite measure.

theorem MeasureTheory.Integrable.measure_lt_lt_top {α : Type u_1} {m : } {μ : } {f : α} (hf : ) {c : } (c_neg : c < 0) :
μ {a : α | f a < c} <

If f is ℝ-valued and integrable, then for any c < 0 the set {x | f x < c} has finite measure.

theorem MeasureTheory.LipschitzWith.integrable_comp_iff_of_antilipschitz {α : Type u_1} {β : Type u_2} {γ : Type u_3} {m : } {μ : } {K : NNReal} {K' : NNReal} {f : αβ} {g : βγ} (hg : ) (hg' : ) (g0 : g 0 = 0) :
theorem MeasureTheory.Integrable.real_toNNReal {α : Type u_1} {m : } {μ : } {f : α} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => (f x).toNNReal) μ
theorem MeasureTheory.ofReal_toReal_ae_eq {α : Type u_1} {m : } {μ : } {f : αENNReal} (hf : ∀ᵐ (x : α) ∂μ, f x < ) :
(fun (x : α) => ENNReal.ofReal (f x).toReal) =ᵐ[μ] f
theorem MeasureTheory.coe_toNNReal_ae_eq {α : Type u_1} {m : } {μ : } {f : αENNReal} (hf : ∀ᵐ (x : α) ∂μ, f x < ) :
(fun (x : α) => (f x).toNNReal) =ᵐ[μ] f
theorem MeasureTheory.hasFiniteIntegral_count_iff {α : Type u_1} {β : Type u_2} {m : } {f : αβ} :
MeasureTheory.HasFiniteIntegral f MeasureTheory.Measure.count Summable fun (x : α) => f x

A function has finite integral for the counting measure iff its norm is summable.

theorem MeasureTheory.integrable_count_iff {α : Type u_1} {β : Type u_2} {m : } {f : αβ} :
MeasureTheory.Integrable f MeasureTheory.Measure.count Summable fun (x : α) => f x

A function is integrable for the counting measure iff its norm is summable.

theorem MeasureTheory.integrable_withDensity_iff_integrable_coe_smul {α : Type u_1} {m : } {μ : } {E : Type u_5} [] {f : αNNReal} (hf : ) {g : αE} :
MeasureTheory.Integrable g (μ.withDensity fun (x : α) => (f x)) MeasureTheory.Integrable (fun (x : α) => (f x) g x) μ
theorem MeasureTheory.integrable_withDensity_iff_integrable_smul {α : Type u_1} {m : } {μ : } {E : Type u_5} [] {f : αNNReal} (hf : ) {g : αE} :
MeasureTheory.Integrable g (μ.withDensity fun (x : α) => (f x)) MeasureTheory.Integrable (fun (x : α) => f x g x) μ
theorem MeasureTheory.integrable_withDensity_iff_integrable_smul' {α : Type u_1} {m : } {μ : } {E : Type u_5} [] {f : αENNReal} (hf : ) (hflt : ∀ᵐ (x : α) ∂μ, f x < ) {g : αE} :
MeasureTheory.Integrable g (μ.withDensity f) MeasureTheory.Integrable (fun (x : α) => (f x).toReal g x) μ
theorem MeasureTheory.integrable_withDensity_iff_integrable_coe_smul₀ {α : Type u_1} {m : } {μ : } {E : Type u_5} [] {f : αNNReal} (hf : ) {g : αE} :
MeasureTheory.Integrable g (μ.withDensity fun (x : α) => (f x)) MeasureTheory.Integrable (fun (x : α) => (f x) g x) μ
theorem MeasureTheory.integrable_withDensity_iff_integrable_smul₀ {α : Type u_1} {m : } {μ : } {E : Type u_5} [] {f : αNNReal} (hf : ) {g : αE} :
MeasureTheory.Integrable g (μ.withDensity fun (x : α) => (f x)) MeasureTheory.Integrable (fun (x : α) => f x g x) μ
theorem MeasureTheory.integrable_withDensity_iff {α : Type u_1} {m : } {μ : } {f : αENNReal} (hf : ) (hflt : ∀ᵐ (x : α) ∂μ, f x < ) {g : α} :
MeasureTheory.Integrable g (μ.withDensity f) MeasureTheory.Integrable (fun (x : α) => g x * (f x).toReal) μ
theorem MeasureTheory.memℒ1_smul_of_L1_withDensity {α : Type u_1} {m : } {μ : } {E : Type u_5} [] {f : αNNReal} (f_meas : ) (u : (MeasureTheory.Lp E 1 (μ.withDensity fun (x : α) => (f x)))) :
MeasureTheory.Memℒp (fun (x : α) => f x u x) 1 μ
noncomputable def MeasureTheory.withDensitySMulLI {α : Type u_1} {m : } (μ : ) {E : Type u_5} [] {f : αNNReal} (f_meas : ) :
(MeasureTheory.Lp E 1 (μ.withDensity fun (x : α) => (f x))) →ₗᵢ[] ()

The map u ↦ f • u is an isometry between the L^1 spaces for μ.withDensity f and μ.

Equations
• One or more equations did not get rendered due to their size.
Instances For
@[simp]
theorem MeasureTheory.withDensitySMulLI_apply {α : Type u_1} {m : } (μ : ) {E : Type u_5} [] {f : αNNReal} (f_meas : ) (u : (MeasureTheory.Lp E 1 (μ.withDensity fun (x : α) => (f x)))) :
() u = MeasureTheory.Memℒp.toLp (fun (x : α) => f x u x)
theorem MeasureTheory.mem_ℒ1_toReal_of_lintegral_ne_top {α : Type u_1} {m : } {μ : } {f : αENNReal} (hfm : ) (hfi : ∫⁻ (x : α), f xμ ) :
MeasureTheory.Memℒp (fun (x : α) => (f x).toReal) 1 μ
theorem MeasureTheory.integrable_toReal_of_lintegral_ne_top {α : Type u_1} {m : } {μ : } {f : αENNReal} (hfm : ) (hfi : ∫⁻ (x : α), f xμ ) :
MeasureTheory.Integrable (fun (x : α) => (f x).toReal) μ

### Lemmas used for defining the positive part of an L¹ function #

theorem MeasureTheory.Integrable.pos_part {α : Type u_1} {m : } {μ : } {f : α} (hf : ) :
MeasureTheory.Integrable (fun (a : α) => max (f a) 0) μ
theorem MeasureTheory.Integrable.neg_part {α : Type u_1} {m : } {μ : } {f : α} (hf : ) :
MeasureTheory.Integrable (fun (a : α) => max (-f a) 0) μ
theorem MeasureTheory.Integrable.smul {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [] (c : 𝕜) {f : αβ} (hf : ) :
theorem IsUnit.integrable_smul_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] {c : 𝕜} (hc : ) (f : αβ) :
theorem MeasureTheory.integrable_smul_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [Module 𝕜 β] [] {c : 𝕜} (hc : c 0) (f : αβ) :
theorem MeasureTheory.Integrable.smul_of_top_right {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] {f : αβ} {φ : α𝕜} (hf : ) (hφ : ) :
theorem MeasureTheory.Integrable.smul_of_top_left {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] {f : αβ} {φ : α𝕜} (hφ : ) (hf : ) :
theorem MeasureTheory.Integrable.smul_const {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] {f : α𝕜} (hf : ) (c : β) :
MeasureTheory.Integrable (fun (x : α) => f x c) μ
theorem MeasureTheory.integrable_smul_const {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {E : Type u_6} [] {f : α𝕜} {c : E} (hc : c 0) :
MeasureTheory.Integrable (fun (x : α) => f x c) μ
theorem MeasureTheory.Integrable.const_mul {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.Integrable (fun (x : α) => c * f x) μ
theorem MeasureTheory.Integrable.const_mul' {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.Integrable ((fun (x : α) => c) * f) μ
theorem MeasureTheory.Integrable.mul_const {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.Integrable (fun (x : α) => f x * c) μ
theorem MeasureTheory.Integrable.mul_const' {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.Integrable (f * fun (x : α) => c) μ
theorem MeasureTheory.integrable_const_mul_iff {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {c : 𝕜} (hc : ) (f : α𝕜) :
MeasureTheory.Integrable (fun (x : α) => c * f x) μ
theorem MeasureTheory.integrable_mul_const_iff {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {c : 𝕜} (hc : ) (f : α𝕜) :
MeasureTheory.Integrable (fun (x : α) => f x * c) μ
theorem MeasureTheory.Integrable.bdd_mul' {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} {g : α𝕜} {c : } (hg : ) (hf : ) (hf_bound : ∀ᵐ (x : α) ∂μ, f x c) :
MeasureTheory.Integrable (fun (x : α) => f x * g x) μ
theorem MeasureTheory.Integrable.div_const {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} {f : α𝕜} (h : ) (c : 𝕜) :
MeasureTheory.Integrable (fun (x : α) => f x / c) μ
theorem MeasureTheory.Integrable.ofReal {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => (f x)) μ
theorem MeasureTheory.Integrable.re_im_iff {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} :
MeasureTheory.Integrable (fun (x : α) => RCLike.re (f x)) μ MeasureTheory.Integrable (fun (x : α) => RCLike.im (f x)) μ
theorem MeasureTheory.Integrable.re {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => RCLike.re (f x)) μ
theorem MeasureTheory.Integrable.im {α : Type u_1} {m : } {μ : } {𝕜 : Type u_5} [] {f : α𝕜} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => RCLike.im (f x)) μ
theorem MeasureTheory.Integrable.trim {α : Type u_1} {m : } {H : Type u_5} {m0 : } {μ' : } {f : αH} (hm : m m0) (hf_int : ) (hf : ) :
MeasureTheory.Integrable f (μ'.trim hm)
theorem MeasureTheory.integrable_of_integrable_trim {α : Type u_1} {m : } {H : Type u_5} {m0 : } {μ' : } {f : αH} (hm : m m0) (hf_int : MeasureTheory.Integrable f (μ'.trim hm)) :
theorem MeasureTheory.integrable_of_forall_fin_meas_le' {α : Type u_1} {m : } {E : Type u_5} {m0 : } {μ : } (hm : m m0) [MeasureTheory.SigmaFinite (μ.trim hm)] (C : ENNReal) (hC : C < ) {f : αE} (hf_meas : ) (hf : ∀ (s : Set α), μ s ∫⁻ (x : α) in s, f x‖₊μ C) :
theorem MeasureTheory.integrable_of_forall_fin_meas_le {α : Type u_1} {m : } {μ : } {E : Type u_5} (C : ENNReal) (hC : C < ) {f : αE} (hf_meas : ) (hf : ∀ (s : Set α), μ s ∫⁻ (x : α) in s, f x‖₊μ C) :

### The predicate Integrable on measurable functions modulo a.e.-equality #

def MeasureTheory.AEEqFun.Integrable {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : α →ₘ[μ] β) :

A class of almost everywhere equal functions is Integrable if its function representative is integrable.

Equations
• f.Integrable =
Instances For
theorem MeasureTheory.AEEqFun.integrable_mk {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
().Integrable
theorem MeasureTheory.AEEqFun.integrable_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : α →ₘ[μ] β} :
f.Integrable
theorem MeasureTheory.AEEqFun.integrable_zero {α : Type u_1} {β : Type u_2} {m : } {μ : } :
theorem MeasureTheory.AEEqFun.Integrable.neg {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : α →ₘ[μ] β} :
f.Integrable(-f).Integrable
theorem MeasureTheory.AEEqFun.integrable_iff_mem_L1 {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : α →ₘ[μ] β} :
f.Integrable f
theorem MeasureTheory.AEEqFun.Integrable.add {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : α →ₘ[μ] β} {g : α →ₘ[μ] β} :
f.Integrableg.Integrable(f + g).Integrable
theorem MeasureTheory.AEEqFun.Integrable.sub {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : α →ₘ[μ] β} {g : α →ₘ[μ] β} (hf : f.Integrable) (hg : g.Integrable) :
(f - g).Integrable
theorem MeasureTheory.AEEqFun.Integrable.smul {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] {c : 𝕜} {f : α →ₘ[μ] β} :
f.Integrable(c f).Integrable
theorem MeasureTheory.L1.integrable_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) :
theorem MeasureTheory.L1.hasFiniteIntegral_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) :
theorem MeasureTheory.L1.stronglyMeasurable_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) :
theorem MeasureTheory.L1.measurable_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } [] [] (f : ()) :
Measurable f
theorem MeasureTheory.L1.aestronglyMeasurable_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) :
theorem MeasureTheory.L1.aemeasurable_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } [] [] (f : ()) :
AEMeasurable (f) μ
theorem MeasureTheory.L1.edist_def {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) (g : ()) :
edist f g = ∫⁻ (a : α), edist (f a) (g a)μ
theorem MeasureTheory.L1.dist_def {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) (g : ()) :
dist f g = (∫⁻ (a : α), edist (f a) (g a)μ).toReal
theorem MeasureTheory.L1.norm_def {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) :
f = (∫⁻ (a : α), f a‖₊μ).toReal
theorem MeasureTheory.L1.norm_sub_eq_lintegral {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) (g : ()) :
f - g = (∫⁻ (x : α), f x - g x‖₊μ).toReal

Computing the norm of a difference between two L¹-functions. Note that this is not a special case of norm_def since (f - g) x and f x - g x are not equal (but only a.e.-equal).

theorem MeasureTheory.L1.ofReal_norm_eq_lintegral {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) :
= ∫⁻ (x : α), f x‖₊μ
theorem MeasureTheory.L1.ofReal_norm_sub_eq_lintegral {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) (g : ()) :
= ∫⁻ (x : α), f x - g x‖₊μ

Computing the norm of a difference between two L¹-functions. Note that this is not a special case of ofReal_norm_eq_lintegral since (f - g) x and f x - g x are not equal (but only a.e.-equal).

def MeasureTheory.Integrable.toL1 {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (hf : ) :
()

Construct the equivalence class [f] of an integrable function f, as a member of the space L1 β 1 μ.

Equations
Instances For
@[simp]
theorem MeasureTheory.Integrable.toL1_coeFn {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : ()) (hf : MeasureTheory.Integrable (f) μ) :
= f
theorem MeasureTheory.Integrable.coeFn_toL1 {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
=ᵐ[μ] f
@[simp]
theorem MeasureTheory.Integrable.toL1_zero {α : Type u_1} {β : Type u_2} {m : } {μ : } (h : ) :
@[simp]
theorem MeasureTheory.Integrable.toL1_eq_mk {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (hf : ) :
@[simp]
theorem MeasureTheory.Integrable.toL1_eq_toL1_iff {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (g : αβ) (hf : ) (hg : ) :
f =ᵐ[μ] g
theorem MeasureTheory.Integrable.toL1_add {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (g : αβ) (hf : ) (hg : ) :
theorem MeasureTheory.Integrable.toL1_neg {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (hf : ) :
theorem MeasureTheory.Integrable.toL1_sub {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (g : αβ) (hf : ) (hg : ) :
theorem MeasureTheory.Integrable.norm_toL1 {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (hf : ) :
= (∫⁻ (a : α), edist (f a) 0μ).toReal
theorem MeasureTheory.Integrable.nnnorm_toL1 {α : Type u_1} {β : Type u_2} {m : } {μ : } {f : αβ} (hf : ) :
= ∫⁻ (a : α), f a‖₊μ
theorem MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (hf : ) :
= (∫⁻ (a : α), μ).toReal
@[simp]
theorem MeasureTheory.Integrable.edist_toL1_toL1 {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (g : αβ) (hf : ) (hg : ) :
= ∫⁻ (a : α), edist (f a) (g a)μ
@[simp]
theorem MeasureTheory.Integrable.edist_toL1_zero {α : Type u_1} {β : Type u_2} {m : } {μ : } (f : αβ) (hf : ) :
= ∫⁻ (a : α), edist (f a) 0μ
theorem MeasureTheory.Integrable.toL1_smul {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] (f : αβ) (hf : ) (k : 𝕜) :
MeasureTheory.Integrable.toL1 (fun (a : α) => k f a) =
theorem MeasureTheory.Integrable.toL1_smul' {α : Type u_1} {β : Type u_2} {m : } {μ : } {𝕜 : Type u_5} [] [Module 𝕜 β] [] (f : αβ) (hf : ) (k : 𝕜) :
=
theorem MeasureTheory.HasFiniteIntegral.restrict {α : Type u_1} {m : } {μ : } {E : Type u_5} {f : αE} (h : ) {s : Set α} :
theorem MeasureTheory.Integrable.restrict {α : Type u_1} {m : } {μ : } {E : Type u_5} {f : αE} (f_intble : ) {s : Set α} :
MeasureTheory.Integrable f (μ.restrict s)
theorem ContinuousLinearMap.integrable_comp {α : Type u_1} {m : } {μ : } {E : Type u_5} {𝕜 : Type u_6} [] {H : Type u_7} [] {φ : αH} (L : H →L[𝕜] E) (φ_int : ) :
MeasureTheory.Integrable (fun (a : α) => L (φ a)) μ
@[simp]
theorem ContinuousLinearEquiv.integrable_comp_iff {α : Type u_1} {m : } {μ : } {E : Type u_5} {𝕜 : Type u_6} [] {H : Type u_7} [] {φ : αH} (L : H ≃L[𝕜] E) :
MeasureTheory.Integrable (fun (a : α) => L (φ a)) μ
@[simp]
theorem LinearIsometryEquiv.integrable_comp_iff {α : Type u_1} {m : } {μ : } {E : Type u_5} {𝕜 : Type u_6} [] {H : Type u_7} [] {φ : αH} (L : H ≃ₗᵢ[𝕜] E) :
MeasureTheory.Integrable (fun (a : α) => L (φ a)) μ
theorem MeasureTheory.Integrable.apply_continuousLinearMap {α : Type u_1} {m : } {μ : } {E : Type u_5} {𝕜 : Type u_6} [] {H : Type u_7} [] {φ : αH →L[𝕜] E} (φ_int : ) (v : H) :
MeasureTheory.Integrable (fun (a : α) => (φ a) v) μ
theorem MeasureTheory.Integrable.fst {α : Type u_1} {m : } {μ : } {E : Type u_5} {F : Type u_6} [] [] {f : αE × F} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => (f x).1) μ
theorem MeasureTheory.Integrable.snd {α : Type u_1} {m : } {μ : } {E : Type u_5} {F : Type u_6} [] [] {f : αE × F} (hf : ) :
MeasureTheory.Integrable (fun (x : α) => (f x).2) μ
theorem MeasureTheory.integrable_prod {α : Type u_1} {m : } {μ : } {E : Type u_5} {F : Type u_6} [] [] {f : αE × F} :
MeasureTheory.Integrable (fun (x : α) => (f x).1) μ MeasureTheory.Integrable (fun (x : α) => (f x).2) μ