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] βis the type ofL¹space, whereαis aMeasureSpaceandβis aNormedAddCommGroupwith aSecondCountableTopology.f : α →ₘ βis a "function" inL¹. In comments,[f]is also used to denote anL¹function.₁can be typed as\1.
Main definitions #
-
Let
f : α → βbe a function, whereαis aMeasureSpaceandβaNormedAddCommGroup. ThenHasFiniteIntegral fmeans(∫⁻ a, ‖f a‖₊) < ∞. -
If
βis moreover aMeasurableSpacethenfis calledIntegrableiffisMeasurableandHasFiniteIntegral fholds.
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
:
theorem
MeasureTheory.lintegral_norm_eq_lintegral_edist
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
:
∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ = ∫⁻ (a : α), edist (f a) 0 ∂μ
theorem
MeasureTheory.lintegral_edist_triangle
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
{h : α → β}
(hf : MeasureTheory.AEStronglyMeasurable f μ)
(hh : MeasureTheory.AEStronglyMeasurable h μ)
:
theorem
MeasureTheory.lintegral_nnnorm_zero
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
:
theorem
MeasureTheory.lintegral_nnnorm_add_left
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
(hf : MeasureTheory.AEStronglyMeasurable f μ)
(g : α → γ)
:
theorem
MeasureTheory.lintegral_nnnorm_add_right
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
(f : α → β)
{g : α → γ}
(hg : MeasureTheory.AEStronglyMeasurable g μ)
:
theorem
MeasureTheory.lintegral_nnnorm_neg
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
:
The predicate HasFiniteIntegral #
def
MeasureTheory.HasFiniteIntegral
{α : Type u_1}
{β : Type u_2}
[NormedAddCommGroup β]
:
{x : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) _auto✝ → Prop
HasFiniteIntegral f μ means that the integral ∫⁻ a, ‖f a‖ ∂μ is finite.
HasFiniteIntegral f means HasFiniteIntegral f volume.
Instances For
theorem
MeasureTheory.hasFiniteIntegral_def
{α : Type u_1}
{β : Type u_2}
[NormedAddCommGroup β]
:
∀ {x : MeasurableSpace α} (f : α → β) (μ : MeasureTheory.Measure α),
MeasureTheory.HasFiniteIntegral f ↔ ∫⁻ (a : α), ↑‖f a‖₊ ∂μ < ⊤
theorem
MeasureTheory.hasFiniteIntegral_iff_norm
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
:
MeasureTheory.HasFiniteIntegral f ↔ ∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ < ⊤
theorem
MeasureTheory.hasFiniteIntegral_iff_edist
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
:
MeasureTheory.HasFiniteIntegral f ↔ ∫⁻ (a : α), edist (f a) 0 ∂μ < ⊤
theorem
MeasureTheory.hasFiniteIntegral_iff_ofReal
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(h : 0 ≤ᶠ[MeasureTheory.Measure.ae μ] f)
:
MeasureTheory.HasFiniteIntegral f ↔ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ < ⊤
theorem
MeasureTheory.hasFiniteIntegral_iff_ofNNReal
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → NNReal}
:
(MeasureTheory.HasFiniteIntegral fun x => ↑(f x)) ↔ ∫⁻ (a : α), ↑(f a) ∂μ < ⊤
theorem
MeasureTheory.HasFiniteIntegral.mono
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(hg : MeasureTheory.HasFiniteIntegral g)
(h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖)
:
theorem
MeasureTheory.HasFiniteIntegral.mono'
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → ℝ}
(hg : MeasureTheory.HasFiniteIntegral g)
(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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(hf : MeasureTheory.HasFiniteIntegral f)
(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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
:
theorem
MeasureTheory.HasFiniteIntegral.congr
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.HasFiniteIntegral f)
(h : f =ᶠ[MeasureTheory.Measure.ae μ] g)
:
theorem
MeasureTheory.hasFiniteIntegral_congr
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(h : f =ᶠ[MeasureTheory.Measure.ae μ] g)
:
theorem
MeasureTheory.hasFiniteIntegral_const_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{c : β}
:
theorem
MeasureTheory.hasFiniteIntegral_const
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasureTheory.IsFiniteMeasure μ]
(c : β)
:
MeasureTheory.HasFiniteIntegral fun x => c
theorem
MeasureTheory.hasFiniteIntegral_of_bounded
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasureTheory.IsFiniteMeasure μ]
{f : α → β}
{C : ℝ}
(hC : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ C)
:
theorem
MeasureTheory.hasFiniteIntegral_of_fintype
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[Fintype α]
[MeasureTheory.IsFiniteMeasure μ]
{f : α → β}
:
theorem
MeasureTheory.HasFiniteIntegral.mono_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.HasFiniteIntegral f)
(hμ : μ ≤ ν)
:
theorem
MeasureTheory.HasFiniteIntegral.add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hμ : MeasureTheory.HasFiniteIntegral f)
(hν : MeasureTheory.HasFiniteIntegral f)
:
theorem
MeasureTheory.HasFiniteIntegral.left_of_add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.HasFiniteIntegral f)
:
theorem
MeasureTheory.HasFiniteIntegral.right_of_add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.HasFiniteIntegral f)
:
@[simp]
theorem
MeasureTheory.hasFiniteIntegral_add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
:
theorem
MeasureTheory.HasFiniteIntegral.smul_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.HasFiniteIntegral f)
{c : ENNReal}
(hc : c ≠ ⊤)
:
@[simp]
theorem
MeasureTheory.hasFiniteIntegral_zero_measure
{α : Type u_1}
{β : Type u_2}
[NormedAddCommGroup β]
{m : MeasurableSpace α}
(f : α → β)
:
@[simp]
theorem
MeasureTheory.hasFiniteIntegral_zero
(α : Type u_1)
(β : Type u_2)
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
[NormedAddCommGroup β]
:
MeasureTheory.HasFiniteIntegral fun x => 0
theorem
MeasureTheory.HasFiniteIntegral.neg
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hfi : MeasureTheory.HasFiniteIntegral f)
:
@[simp]
theorem
MeasureTheory.hasFiniteIntegral_neg_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
:
theorem
MeasureTheory.HasFiniteIntegral.norm
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hfi : MeasureTheory.HasFiniteIntegral f)
:
MeasureTheory.HasFiniteIntegral fun a => ‖f a‖
theorem
MeasureTheory.hasFiniteIntegral_norm_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
:
(MeasureTheory.HasFiniteIntegral fun a => ‖f a‖) ↔ MeasureTheory.HasFiniteIntegral f
theorem
MeasureTheory.hasFiniteIntegral_toReal_of_lintegral_ne_top
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ENNReal}
(hf : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
:
MeasureTheory.HasFiniteIntegral fun x => ENNReal.toReal (f x)
theorem
MeasureTheory.isFiniteMeasure_withDensity_ofReal
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hfi : MeasureTheory.HasFiniteIntegral f)
:
MeasureTheory.IsFiniteMeasure (MeasureTheory.Measure.withDensity μ fun x => ENNReal.ofReal (f x))
theorem
MeasureTheory.all_ae_ofReal_F_le_bound
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{F : ℕ → α → β}
{bound : α → ℝ}
(h : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound a)
(n : 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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{F : ℕ → α → β}
{f : α → β}
(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 (ENNReal.ofReal ‖f a‖))
theorem
MeasureTheory.all_ae_ofReal_f_le_bound
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{F : ℕ → α → β}
{f : α → β}
{bound : α → ℝ}
(h_bound : ∀ (n : 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 ‖f a‖ ≤ ENNReal.ofReal (bound a)
theorem
MeasureTheory.hasFiniteIntegral_of_dominated_convergence
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{F : ℕ → α → β}
{f : α → β}
{bound : α → ℝ}
(bound_hasFiniteIntegral : MeasureTheory.HasFiniteIntegral bound)
(h_bound : ∀ (n : 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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{F : ℕ → α → β}
{f : α → β}
{bound : α → ℝ}
(F_measurable : ∀ (n : ℕ), MeasureTheory.AEStronglyMeasurable (F n) μ)
(bound_hasFiniteIntegral : MeasureTheory.HasFiniteIntegral bound)
(h_bound : ∀ (n : 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 a‖ ∂μ) 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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.HasFiniteIntegral f)
:
MeasureTheory.HasFiniteIntegral fun a => max (f a) 0
theorem
MeasureTheory.HasFiniteIntegral.min_zero
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.HasFiniteIntegral f)
:
MeasureTheory.HasFiniteIntegral fun a => min (f a) 0
theorem
MeasureTheory.HasFiniteIntegral.smul
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedAddCommGroup 𝕜]
[SMulZeroClass 𝕜 β]
[BoundedSMul 𝕜 β]
(c : 𝕜)
{f : α → β}
:
theorem
MeasureTheory.hasFiniteIntegral_smul_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[MulActionWithZero 𝕜 β]
[BoundedSMul 𝕜 β]
{c : 𝕜}
(hc : IsUnit c)
(f : α → β)
:
theorem
MeasureTheory.HasFiniteIntegral.const_mul
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.HasFiniteIntegral f)
(c : 𝕜)
:
MeasureTheory.HasFiniteIntegral fun x => c * f x
theorem
MeasureTheory.HasFiniteIntegral.mul_const
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.HasFiniteIntegral f)
(c : 𝕜)
:
MeasureTheory.HasFiniteIntegral fun x => f x * c
The predicate Integrable #
def
MeasureTheory.Integrable
{β : Type u_2}
[NormedAddCommGroup β]
{α : Type u_5}
:
{x : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) _auto✝ → Prop
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.integrable_def
{β : Type u_2}
[NormedAddCommGroup β]
{α : Type u_5}
:
∀ {x : MeasurableSpace α} (f : α → β) (μ : MeasureTheory.Measure α),
MeasureTheory.Integrable f ↔ MeasureTheory.AEStronglyMeasurable f μ ∧ MeasureTheory.HasFiniteIntegral f
theorem
MeasureTheory.memℒp_one_iff_integrable
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
:
theorem
MeasureTheory.Integrable.aestronglyMeasurable
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.Integrable.aemeasurable
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasurableSpace β]
[BorelSpace β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.Integrable.hasFiniteIntegral
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.Integrable.mono
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(hg : MeasureTheory.Integrable g)
(hf : MeasureTheory.AEStronglyMeasurable f μ)
(h : ∀ᵐ (a : α) ∂μ, ‖f a‖ ≤ ‖g a‖)
:
theorem
MeasureTheory.Integrable.mono'
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → ℝ}
(hg : MeasureTheory.Integrable g)
(hf : MeasureTheory.AEStronglyMeasurable f μ)
(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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.AEStronglyMeasurable g μ)
(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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(hf : MeasureTheory.AEStronglyMeasurable f μ)
(hg : MeasureTheory.AEStronglyMeasurable g μ)
(h : ∀ᵐ (a : α) ∂μ, ‖f a‖ = ‖g a‖)
:
theorem
MeasureTheory.Integrable.congr
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(h : f =ᶠ[MeasureTheory.Measure.ae μ] g)
:
theorem
MeasureTheory.integrable_congr
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(h : f =ᶠ[MeasureTheory.Measure.ae μ] g)
:
theorem
MeasureTheory.integrable_const_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{c : β}
:
@[simp]
theorem
MeasureTheory.integrable_const
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasureTheory.IsFiniteMeasure μ]
(c : β)
:
MeasureTheory.Integrable fun x => c
@[simp]
theorem
MeasureTheory.integrable_of_fintype
{α : Type u_1}
{β : Type u_2}
[NormedAddCommGroup β]
[Fintype α]
[MeasurableSpace α]
[MeasurableSingletonClass α]
(μ : MeasureTheory.Measure α)
[MeasureTheory.IsFiniteMeasure μ]
(f : α → β)
:
MeasureTheory.Integrable fun a => f a
theorem
MeasureTheory.Memℒp.integrable_norm_rpow
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{p : ENNReal}
(hf : MeasureTheory.Memℒp f p)
(hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ⊤)
:
MeasureTheory.Integrable fun x => ‖f x‖ ^ ENNReal.toReal p
theorem
MeasureTheory.Memℒp.integrable_norm_rpow'
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasureTheory.IsFiniteMeasure μ]
{f : α → β}
{p : ENNReal}
(hf : MeasureTheory.Memℒp f p)
:
MeasureTheory.Integrable fun x => ‖f x‖ ^ ENNReal.toReal p
theorem
MeasureTheory.Integrable.mono_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.Integrable f)
(hμ : μ ≤ ν)
:
theorem
MeasureTheory.Integrable.of_measure_le_smul
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{μ' : MeasureTheory.Measure α}
(c : ENNReal)
(hc : c ≠ ⊤)
(hμ'_le : μ' ≤ c • μ)
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.Integrable.add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hμ : MeasureTheory.Integrable f)
(hν : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.Integrable.left_of_add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.Integrable.right_of_add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.Integrable f)
:
@[simp]
theorem
MeasureTheory.integrable_add_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{ν : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
:
@[simp]
theorem
MeasureTheory.integrable_zero_measure
{α : Type u_1}
{β : Type u_2}
[NormedAddCommGroup β]
:
∀ {x : MeasurableSpace α} {f : α → β}, MeasureTheory.Integrable f
theorem
MeasureTheory.integrable_finset_sum_measure
{α : Type u_1}
{β : Type u_2}
[NormedAddCommGroup β]
{ι : Type u_5}
{m : MeasurableSpace α}
{f : α → β}
{μ : ι → MeasureTheory.Measure α}
{s : Finset ι}
:
MeasureTheory.Integrable f ↔ ∀ (i : ι), i ∈ s → MeasureTheory.Integrable f
theorem
MeasureTheory.Integrable.smul_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.Integrable f)
{c : ENNReal}
(hc : c ≠ ⊤)
:
theorem
MeasureTheory.integrable_smul_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{c : ENNReal}
(h₁ : c ≠ 0)
(h₂ : c ≠ ⊤)
:
theorem
MeasureTheory.integrable_inv_smul_measure
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{c : ENNReal}
(h₁ : c ≠ 0)
(h₂ : c ≠ ⊤)
:
theorem
MeasureTheory.Integrable.to_average
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(h : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.integrable_average
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasureTheory.IsFiniteMeasure μ]
{f : α → β}
:
theorem
MeasureTheory.integrable_map_measure
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
{f : α → δ}
{g : δ → β}
(hg : MeasureTheory.AEStronglyMeasurable g (MeasureTheory.Measure.map f μ))
(hf : AEMeasurable f)
:
theorem
MeasureTheory.Integrable.comp_aemeasurable
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
{f : α → δ}
{g : δ → β}
(hg : MeasureTheory.Integrable g)
(hf : AEMeasurable f)
:
MeasureTheory.Integrable (g ∘ f)
theorem
MeasureTheory.Integrable.comp_measurable
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
{f : α → δ}
{g : δ → β}
(hg : MeasureTheory.Integrable g)
(hf : Measurable f)
:
MeasureTheory.Integrable (g ∘ f)
theorem
MeasurableEmbedding.integrable_map_iff
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
{f : α → δ}
(hf : MeasurableEmbedding f)
{g : δ → β}
:
theorem
MeasureTheory.integrable_map_equiv
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
(f : α ≃ᵐ δ)
(g : δ → β)
:
theorem
MeasureTheory.MeasurePreserving.integrable_comp
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
{ν : MeasureTheory.Measure δ}
{g : δ → β}
{f : α → δ}
(hf : MeasureTheory.MeasurePreserving f)
(hg : MeasureTheory.AEStronglyMeasurable g ν)
:
theorem
MeasureTheory.MeasurePreserving.integrable_comp_emb
{α : Type u_1}
{β : Type u_2}
{δ : Type u_4}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[MeasurableSpace δ]
[NormedAddCommGroup β]
{f : α → δ}
{ν : MeasureTheory.Measure δ}
(h₁ : MeasureTheory.MeasurePreserving f)
(h₂ : MeasurableEmbedding f)
{g : δ → β}
:
theorem
MeasureTheory.lintegral_edist_lt_top
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
@[simp]
theorem
MeasureTheory.integrable_zero
(α : Type u_1)
(β : Type u_2)
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
[NormedAddCommGroup β]
:
MeasureTheory.Integrable fun x => 0
theorem
MeasureTheory.Integrable.add'
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
theorem
MeasureTheory.Integrable.add
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable (f + g)
theorem
MeasureTheory.integrable_finset_sum'
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{ι : Type u_5}
(s : Finset ι)
{f : ι → α → β}
(hf : ∀ (i : ι), i ∈ s → MeasureTheory.Integrable (f i))
:
MeasureTheory.Integrable (Finset.sum s fun i => f i)
theorem
MeasureTheory.integrable_finset_sum
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{ι : Type u_5}
(s : Finset ι)
{f : ι → α → β}
(hf : ∀ (i : ι), i ∈ s → MeasureTheory.Integrable (f i))
:
MeasureTheory.Integrable fun a => Finset.sum s fun i => f i a
theorem
MeasureTheory.Integrable.neg
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
@[simp]
theorem
MeasureTheory.integrable_neg_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
:
theorem
MeasureTheory.Integrable.sub
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable (f - g)
theorem
MeasureTheory.Integrable.norm
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun a => ‖f a‖
theorem
MeasureTheory.Integrable.inf
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{β : Type u_5}
[NormedLatticeAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable (f ⊓ g)
theorem
MeasureTheory.Integrable.sup
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{β : Type u_5}
[NormedLatticeAddCommGroup β]
{f : α → β}
{g : α → β}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable (f ⊔ g)
theorem
MeasureTheory.Integrable.abs
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{β : Type u_5}
[NormedLatticeAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun a => |f a|
theorem
MeasureTheory.Integrable.bdd_mul
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{F : Type u_5}
[NormedDivisionRing F]
{f : α → F}
{g : α → F}
(hint : MeasureTheory.Integrable g)
(hm : MeasureTheory.AEStronglyMeasurable f μ)
(hfbdd : ∃ C, ∀ (x : α), ‖f x‖ ≤ C)
:
MeasureTheory.Integrable fun x => f x * g x
theorem
MeasureTheory.Integrable.essSup_smul
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedField 𝕜]
[NormedSpace 𝕜 β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
{g : α → 𝕜}
(g_aestronglyMeasurable : MeasureTheory.AEStronglyMeasurable g μ)
(ess_sup_g : essSup (fun x => ↑‖g x‖₊) μ ≠ ⊤)
:
MeasureTheory.Integrable fun x => g x • f x
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{f : α → 𝕜}
(hf : MeasureTheory.Integrable f)
{g : α → β}
(g_aestronglyMeasurable : MeasureTheory.AEStronglyMeasurable g μ)
(ess_sup_g : essSup (fun x => ↑‖g x‖₊) μ ≠ ⊤)
:
MeasureTheory.Integrable fun x => f x • g x
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.AEStronglyMeasurable f μ)
:
(MeasureTheory.Integrable fun a => ‖f a‖) ↔ MeasureTheory.Integrable f
theorem
MeasureTheory.integrable_of_norm_sub_le
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f₀ : α → β}
{f₁ : α → β}
{g : α → ℝ}
(hf₁_m : MeasureTheory.AEStronglyMeasurable f₁ μ)
(hf₀_i : MeasureTheory.Integrable f₀)
(hg_i : MeasureTheory.Integrable g)
(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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{f : α → β}
{g : α → γ}
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable fun x => (f x, g x)
theorem
MeasureTheory.Memℒp.integrable
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{q : ENNReal}
(hq1 : 1 ≤ q)
{f : α → β}
[MeasureTheory.IsFiniteMeasure μ]
(hfq : MeasureTheory.Memℒp f q)
:
theorem
MeasureTheory.Integrable.measure_norm_ge_lt_top
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
{ε : ℝ}
(hε : 0 < ε)
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
{ε : ℝ}
(hε : 0 < ε)
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
{ε : ℝ}
(ε_pos : 0 < ε)
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
{c : ℝ}
(c_neg : c < 0)
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
{ε : ℝ}
(ε_pos : 0 < ε)
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
{c : ℝ}
(c_neg : c < 0)
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[NormedAddCommGroup γ]
{K : NNReal}
{K' : NNReal}
{f : α → β}
{g : β → γ}
(hg : LipschitzWith K g)
(hg' : AntilipschitzWith K' g)
(g0 : g 0 = 0)
:
theorem
MeasureTheory.Integrable.real_toNNReal
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun x => ↑(Real.toNNReal (f x))
theorem
MeasureTheory.ofReal_toReal_ae_eq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ENNReal}
(hf : ∀ᵐ (x : α) ∂μ, f x < ⊤)
:
(fun x => ENNReal.ofReal (ENNReal.toReal (f x))) =ᶠ[MeasureTheory.Measure.ae μ] f
theorem
MeasureTheory.coe_toNNReal_ae_eq
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ENNReal}
(hf : ∀ᵐ (x : α) ∂μ, f x < ⊤)
:
(fun x => ↑(ENNReal.toNNReal (f x))) =ᶠ[MeasureTheory.Measure.ae μ] f
theorem
MeasureTheory.integrable_withDensity_iff_integrable_coe_smul
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(hf : Measurable f)
{g : α → E}
:
MeasureTheory.Integrable g ↔ MeasureTheory.Integrable fun x => ↑(f x) • g x
theorem
MeasureTheory.integrable_withDensity_iff_integrable_smul
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(hf : Measurable f)
{g : α → E}
:
MeasureTheory.Integrable g ↔ MeasureTheory.Integrable fun x => f x • g x
theorem
MeasureTheory.integrable_withDensity_iff_integrable_smul'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → ENNReal}
(hf : Measurable f)
(hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤)
{g : α → E}
:
MeasureTheory.Integrable g ↔ MeasureTheory.Integrable fun x => ENNReal.toReal (f x) • g x
theorem
MeasureTheory.integrable_withDensity_iff_integrable_coe_smul₀
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(hf : AEMeasurable f)
{g : α → E}
:
MeasureTheory.Integrable g ↔ MeasureTheory.Integrable fun x => ↑(f x) • g x
theorem
MeasureTheory.integrable_withDensity_iff_integrable_smul₀
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(hf : AEMeasurable f)
{g : α → E}
:
MeasureTheory.Integrable g ↔ MeasureTheory.Integrable fun x => f x • g x
theorem
MeasureTheory.integrable_withDensity_iff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ENNReal}
(hf : Measurable f)
(hflt : ∀ᵐ (x : α) ∂μ, f x < ⊤)
{g : α → ℝ}
:
MeasureTheory.Integrable g ↔ MeasureTheory.Integrable fun x => g x * ENNReal.toReal (f x)
theorem
MeasureTheory.memℒ1_smul_of_L1_withDensity
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(f_meas : Measurable f)
(u : { x // x ∈ MeasureTheory.Lp E 1 })
:
MeasureTheory.Memℒp (fun x => f x • ↑↑u x) 1
noncomputable def
MeasureTheory.withDensitySMulLI
{α : Type u_1}
{m : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(f_meas : Measurable f)
:
{ x // x ∈ MeasureTheory.Lp E 1 } →ₗᵢ[ℝ] { x // x ∈ MeasureTheory.Lp E 1 }
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 : MeasurableSpace α}
(μ : MeasureTheory.Measure α)
{E : Type u_5}
[NormedAddCommGroup E]
[NormedSpace ℝ E]
{f : α → NNReal}
(f_meas : Measurable f)
(u : { x // x ∈ MeasureTheory.Lp E 1 })
:
↑(MeasureTheory.withDensitySMulLI μ f_meas) u = MeasureTheory.Memℒp.toLp (fun x => f x • ↑↑u x) (_ : MeasureTheory.Memℒp (fun x => f x • ↑↑u x) 1)
theorem
MeasureTheory.mem_ℒ1_toReal_of_lintegral_ne_top
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ENNReal}
(hfm : AEMeasurable f)
(hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
:
MeasureTheory.Memℒp (fun x => ENNReal.toReal (f x)) 1
theorem
MeasureTheory.integrable_toReal_of_lintegral_ne_top
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ENNReal}
(hfm : AEMeasurable f)
(hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
:
MeasureTheory.Integrable fun x => ENNReal.toReal (f x)
Lemmas used for defining the positive part of an L¹ function #
theorem
MeasureTheory.Integrable.pos_part
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun a => max (f a) 0
theorem
MeasureTheory.Integrable.neg_part
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun a => max (-f a) 0
theorem
MeasureTheory.Integrable.smul
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedAddCommGroup 𝕜]
[SMulZeroClass 𝕜 β]
[BoundedSMul 𝕜 β]
(c : 𝕜)
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable (c • f)
theorem
IsUnit.integrable_smul_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{c : 𝕜}
(hc : IsUnit c)
(f : α → β)
:
theorem
MeasureTheory.integrable_smul_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedDivisionRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{c : 𝕜}
(hc : c ≠ 0)
(f : α → β)
:
theorem
MeasureTheory.Integrable.smul_of_top_right
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{f : α → β}
{φ : α → 𝕜}
(hf : MeasureTheory.Integrable f)
(hφ : MeasureTheory.Memℒp φ ⊤)
:
MeasureTheory.Integrable (φ • f)
theorem
MeasureTheory.Integrable.smul_of_top_left
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{f : α → β}
{φ : α → 𝕜}
(hφ : MeasureTheory.Integrable φ)
(hf : MeasureTheory.Memℒp f ⊤)
:
MeasureTheory.Integrable (φ • f)
theorem
MeasureTheory.Integrable.smul_const
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{f : α → 𝕜}
(hf : MeasureTheory.Integrable f)
(c : β)
:
MeasureTheory.Integrable fun x => f x • c
theorem
MeasureTheory.integrable_smul_const
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NontriviallyNormedField 𝕜]
[CompleteSpace 𝕜]
{E : Type u_6}
[NormedAddCommGroup E]
[NormedSpace 𝕜 E]
{f : α → 𝕜}
{c : E}
(hc : c ≠ 0)
:
(MeasureTheory.Integrable fun x => f x • c) ↔ MeasureTheory.Integrable f
theorem
MeasureTheory.Integrable.const_mul
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.Integrable f)
(c : 𝕜)
:
MeasureTheory.Integrable fun x => c * f x
theorem
MeasureTheory.Integrable.const_mul'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.Integrable f)
(c : 𝕜)
:
MeasureTheory.Integrable ((fun x => c) * f)
theorem
MeasureTheory.Integrable.mul_const
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.Integrable f)
(c : 𝕜)
:
MeasureTheory.Integrable fun x => f x * c
theorem
MeasureTheory.Integrable.mul_const'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.Integrable f)
(c : 𝕜)
:
MeasureTheory.Integrable (f * fun x => c)
theorem
MeasureTheory.integrable_const_mul_iff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{c : 𝕜}
(hc : IsUnit c)
(f : α → 𝕜)
:
(MeasureTheory.Integrable fun x => c * f x) ↔ MeasureTheory.Integrable f
theorem
MeasureTheory.integrable_mul_const_iff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{c : 𝕜}
(hc : IsUnit c)
(f : α → 𝕜)
:
(MeasureTheory.Integrable fun x => f x * c) ↔ MeasureTheory.Integrable f
theorem
MeasureTheory.Integrable.bdd_mul'
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedRing 𝕜]
{f : α → 𝕜}
{g : α → 𝕜}
{c : ℝ}
(hg : MeasureTheory.Integrable g)
(hf : MeasureTheory.AEStronglyMeasurable f μ)
(hf_bound : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ c)
:
MeasureTheory.Integrable fun x => f x * g x
theorem
MeasureTheory.Integrable.div_const
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[NormedDivisionRing 𝕜]
{f : α → 𝕜}
(h : MeasureTheory.Integrable f)
(c : 𝕜)
:
MeasureTheory.Integrable fun x => f x / c
theorem
MeasureTheory.Integrable.ofReal
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[IsROrC 𝕜]
{f : α → ℝ}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun x => ↑(f x)
theorem
MeasureTheory.Integrable.re_im_iff
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[IsROrC 𝕜]
{f : α → 𝕜}
:
((MeasureTheory.Integrable fun x => ↑IsROrC.re (f x)) ∧ MeasureTheory.Integrable fun x => ↑IsROrC.im (f x)) ↔ MeasureTheory.Integrable f
theorem
MeasureTheory.Integrable.re
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[IsROrC 𝕜]
{f : α → 𝕜}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun x => ↑IsROrC.re (f x)
theorem
MeasureTheory.Integrable.im
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{𝕜 : Type u_5}
[IsROrC 𝕜]
{f : α → 𝕜}
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable fun x => ↑IsROrC.im (f x)
theorem
MeasureTheory.Integrable.trim
{α : Type u_1}
{m : MeasurableSpace α}
{H : Type u_5}
[NormedAddCommGroup H]
{m0 : MeasurableSpace α}
{μ' : MeasureTheory.Measure α}
{f : α → H}
(hm : m ≤ m0)
(hf_int : MeasureTheory.Integrable f)
(hf : MeasureTheory.StronglyMeasurable f)
:
theorem
MeasureTheory.integrable_of_integrable_trim
{α : Type u_1}
{m : MeasurableSpace α}
{H : Type u_5}
[NormedAddCommGroup H]
{m0 : MeasurableSpace α}
{μ' : MeasureTheory.Measure α}
{f : α → H}
(hm : m ≤ m0)
(hf_int : MeasureTheory.Integrable f)
:
theorem
MeasureTheory.integrable_of_forall_fin_meas_le'
{α : Type u_1}
{m : MeasurableSpace α}
{E : Type u_5}
{m0 : MeasurableSpace α}
[NormedAddCommGroup E]
{μ : MeasureTheory.Measure α}
(hm : m ≤ m0)
[MeasureTheory.SigmaFinite (MeasureTheory.Measure.trim μ hm)]
(C : ENNReal)
(hC : C < ⊤)
{f : α → E}
(hf_meas : MeasureTheory.AEStronglyMeasurable f μ)
(hf : ∀ (s : Set α), MeasurableSet s → ↑↑μ s ≠ ⊤ → ∫⁻ (x : α) in s, ↑‖f x‖₊ ∂μ ≤ C)
:
theorem
MeasureTheory.integrable_of_forall_fin_meas_le
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
[MeasureTheory.SigmaFinite μ]
(C : ENNReal)
(hC : C < ⊤)
{f : α → E}
(hf_meas : MeasureTheory.AEStronglyMeasurable f μ)
(hf : ∀ (s : Set α), MeasurableSet s → ↑↑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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α →ₘ[μ] β)
:
A class of almost everywhere equal functions is Integrable if its function representative
is integrable.
Equations
Instances For
theorem
MeasureTheory.AEEqFun.integrable_mk
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.AEStronglyMeasurable f μ)
:
theorem
MeasureTheory.AEEqFun.integrable_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α →ₘ[μ] β}
:
theorem
MeasureTheory.AEEqFun.integrable_zero
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
:
theorem
MeasureTheory.AEEqFun.Integrable.neg
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α →ₘ[μ] β}
:
theorem
MeasureTheory.AEEqFun.integrable_iff_mem_L1
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α →ₘ[μ] β}
:
theorem
MeasureTheory.AEEqFun.Integrable.add
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α →ₘ[μ] β}
{g : α →ₘ[μ] β}
:
theorem
MeasureTheory.AEEqFun.Integrable.sub
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α →ₘ[μ] β}
{g : α →ₘ[μ] β}
(hf : MeasureTheory.AEEqFun.Integrable f)
(hg : MeasureTheory.AEEqFun.Integrable g)
:
theorem
MeasureTheory.AEEqFun.Integrable.smul
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
{c : 𝕜}
{f : α →ₘ[μ] β}
:
theorem
MeasureTheory.L1.integrable_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.hasFiniteIntegral_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.stronglyMeasurable_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.measurable_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasurableSpace β]
[BorelSpace β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
Measurable ↑↑f
theorem
MeasureTheory.L1.aestronglyMeasurable_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.aemeasurable_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
[MeasurableSpace β]
[BorelSpace β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
AEMeasurable ↑↑f
theorem
MeasureTheory.L1.edist_def
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
(g : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.dist_def
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
(g : { x // x ∈ MeasureTheory.Lp β 1 })
:
dist f g = ENNReal.toReal (∫⁻ (a : α), edist (↑↑f a) (↑↑g a) ∂μ)
theorem
MeasureTheory.L1.norm_def
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.norm_sub_eq_lintegral
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
(g : { x // x ∈ MeasureTheory.Lp β 1 })
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
:
theorem
MeasureTheory.L1.ofReal_norm_sub_eq_lintegral
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
(g : { x // x ∈ MeasureTheory.Lp β 1 })
:
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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
:
{ x // x ∈ MeasureTheory.Lp β 1 }
Construct the equivalence class [f] of an integrable function f, as a member of the
space L1 β 1 μ.
Equations
- MeasureTheory.Integrable.toL1 f hf = MeasureTheory.Memℒp.toLp f (_ : MeasureTheory.Memℒp f 1)
Instances For
@[simp]
theorem
MeasureTheory.Integrable.toL1_coeFn
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : { x // x ∈ MeasureTheory.Lp β 1 })
(hf : MeasureTheory.Integrable ↑↑f)
:
MeasureTheory.Integrable.toL1 (↑↑f) hf = f
theorem
MeasureTheory.Integrable.coeFn_toL1
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{f : α → β}
(hf : MeasureTheory.Integrable f)
:
@[simp]
theorem
MeasureTheory.Integrable.toL1_zero
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(h : MeasureTheory.Integrable 0)
:
@[simp]
theorem
MeasureTheory.Integrable.toL1_eq_mk
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
:
↑(MeasureTheory.Integrable.toL1 f hf) = MeasureTheory.AEEqFun.mk f (_ : MeasureTheory.AEStronglyMeasurable f μ)
@[simp]
theorem
MeasureTheory.Integrable.toL1_eq_toL1_iff
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(g : α → β)
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
theorem
MeasureTheory.Integrable.toL1_add
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(g : α → β)
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable.toL1 (f + g) (_ : MeasureTheory.Integrable (f + g)) = MeasureTheory.Integrable.toL1 f hf + MeasureTheory.Integrable.toL1 g hg
theorem
MeasureTheory.Integrable.toL1_neg
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
:
MeasureTheory.Integrable.toL1 (-f) (_ : MeasureTheory.Integrable (-f)) = -MeasureTheory.Integrable.toL1 f hf
theorem
MeasureTheory.Integrable.toL1_sub
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(g : α → β)
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
MeasureTheory.Integrable.toL1 (f - g) (_ : MeasureTheory.Integrable (f - g)) = MeasureTheory.Integrable.toL1 f hf - MeasureTheory.Integrable.toL1 g hg
theorem
MeasureTheory.Integrable.norm_toL1
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
:
‖MeasureTheory.Integrable.toL1 f hf‖ = ENNReal.toReal (∫⁻ (a : α), edist (f a) 0 ∂μ)
theorem
MeasureTheory.Integrable.norm_toL1_eq_lintegral_norm
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
:
‖MeasureTheory.Integrable.toL1 f hf‖ = ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ)
@[simp]
theorem
MeasureTheory.Integrable.edist_toL1_toL1
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(g : α → β)
(hf : MeasureTheory.Integrable f)
(hg : MeasureTheory.Integrable g)
:
edist (MeasureTheory.Integrable.toL1 f hf) (MeasureTheory.Integrable.toL1 g 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 : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
:
edist (MeasureTheory.Integrable.toL1 f hf) 0 = ∫⁻ (a : α), edist (f a) 0 ∂μ
theorem
MeasureTheory.Integrable.toL1_smul
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
(k : 𝕜)
:
MeasureTheory.Integrable.toL1 (fun a => k • f a) (_ : MeasureTheory.Integrable (k • f)) = k • MeasureTheory.Integrable.toL1 f hf
theorem
MeasureTheory.Integrable.toL1_smul'
{α : Type u_1}
{β : Type u_2}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
[NormedAddCommGroup β]
{𝕜 : Type u_5}
[NormedRing 𝕜]
[Module 𝕜 β]
[BoundedSMul 𝕜 β]
(f : α → β)
(hf : MeasureTheory.Integrable f)
(k : 𝕜)
:
MeasureTheory.Integrable.toL1 (k • f) (_ : MeasureTheory.Integrable (k • f)) = k • MeasureTheory.Integrable.toL1 f hf
theorem
MeasureTheory.HasFiniteIntegral.restrict
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
{f : α → E}
(h : MeasureTheory.HasFiniteIntegral f)
{s : Set α}
:
theorem
MeasureTheory.Integrable.restrict
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
{f : α → E}
(f_intble : MeasureTheory.Integrable f)
{s : Set α}
:
theorem
MeasureTheory.Integrable.apply_continuousLinearMap
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
{𝕜 : Type u_6}
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{H : Type u_7}
[NormedAddCommGroup H]
[NormedSpace 𝕜 H]
{φ : α → H →L[𝕜] E}
(φ_int : MeasureTheory.Integrable φ)
(v : H)
:
MeasureTheory.Integrable fun a => ↑(φ a) v
theorem
ContinuousLinearMap.integrable_comp
{α : Type u_1}
{m : MeasurableSpace α}
{μ : MeasureTheory.Measure α}
{E : Type u_5}
[NormedAddCommGroup E]
{𝕜 : Type u_6}
[NontriviallyNormedField 𝕜]
[NormedSpace 𝕜 E]
{H : Type u_7}
[NormedAddCommGroup H]
[NormedSpace 𝕜 H]
{φ : α → H}
(L : H →L[𝕜] E)
(φ_int : MeasureTheory.Integrable φ)
:
MeasureTheory.Integrable fun a => ↑L (φ a)