Lower Lebesgue integral for ℝ≥0∞-valued functions #
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We define the lower Lebesgue integral of an ℝ≥0∞-valued function.
Notation #
We introduce the following notation for the lower Lebesgue integral of a function f : α → ℝ≥0∞.
∫⁻ x, f x ∂μ: integral of a functionf : α → ℝ≥0∞with respect to a measureμ;∫⁻ x, f x: integral of a functionf : α → ℝ≥0∞with respect to the canonical measurevolumeonα;∫⁻ x in s, f x ∂μ: integral of a functionf : α → ℝ≥0∞over a setswith respect to a measureμ, defined as∫⁻ x, f x ∂(μ.restrict s);∫⁻ x in s, f x: integral of a functionf : α → ℝ≥0∞over a setswith respect to the canonical measurevolume, defined as∫⁻ x, f x ∂(volume.restrict s).
theorem
measure_theory.restrict_dirac'
{α : Type u_1}
{mα : measurable_space α}
{a : α}
{s : set α}
(hs : measurable_set s)
[decidable (a ∈ s)] :
(measure_theory.measure.dirac a).restrict s = ite (a ∈ s) (measure_theory.measure.dirac a) 0
theorem
measure_theory.restrict_dirac
{α : Type u_1}
{mα : measurable_space α}
{a : α}
{s : set α}
[measurable_singleton_class α]
[decidable (a ∈ s)] :
(measure_theory.measure.dirac a).restrict s = ite (a ∈ s) (measure_theory.measure.dirac a) 0
@[irreducible]
noncomputable
def
measure_theory.lintegral
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
(f : α → ennreal) :
The lower Lebesgue integral of a function f with respect to a measure μ.
Equations
- measure_theory.lintegral μ f = ⨆ (g : measure_theory.simple_func α ennreal) (hf : ⇑g ≤ f), g.lintegral μ
In the notation for integrals, an expression like ∫⁻ x, g ‖x‖ ∂μ will not be parsed correctly,
and needs parentheses. We do not set the binding power of r to 0, because then
∫⁻ x, f x = 0 will be parsed incorrectly.
theorem
measure_theory.simple_func.lintegral_eq_lintegral
{α : Type u_1}
{m : measurable_space α}
(f : measure_theory.simple_func α ennreal)
(μ : measure_theory.measure α) :
theorem
measure_theory.lintegral_mono
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
⦃f g : α → ennreal⦄
(hfg : f ≤ g) :
theorem
measure_theory.supr_lintegral_measurable_le_eq_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal) :
theorem
measure_theory.monotone_lintegral
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α) :
@[simp]
theorem
measure_theory.lintegral_const
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(c : ennreal) :
theorem
measure_theory.lintegral_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.lintegral_zero_fun
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
measure_theory.lintegral μ 0 = 0
@[simp]
theorem
measure_theory.lintegral_one
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.set_lintegral_const
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : set α)
(c : ennreal) :
theorem
measure_theory.set_lintegral_one
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : set α) :
theorem
measure_theory.set_lintegral_const_lt_top
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
(s : set α)
{c : ennreal}
(hc : c ≠ ⊤) :
theorem
measure_theory.lintegral_const_lt_top
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.is_finite_measure μ]
{c : ennreal}
(hc : c ≠ ⊤) :
theorem
measure_theory.exists_measurable_le_lintegral_eq
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
(f : α → ennreal) :
For any function f : α → ℝ≥0∞, there exists a measurable function g ≤ f with the same
integral.
theorem
measure_theory.lintegral_eq_nnreal
{α : Type u_1}
{m : measurable_space α}
(f : α → ennreal)
(μ : measure_theory.measure α) :
∫⁻ a in s, f a ∂μ is defined as the supremum of integrals of simple functions
φ : α →ₛ ℝ≥0∞ such that φ ≤ f. This lemma says that it suffices to take
functions φ : α →ₛ ℝ≥0.
theorem
measure_theory.exists_simple_func_forall_lintegral_sub_lt_of_pos
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(h : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
{ε : ennreal}
(hε : ε ≠ 0) :
theorem
measure_theory.set_lintegral_mono_ae
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g)
(hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x ≤ g x) :
theorem
measure_theory.set_lintegral_mono
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g)
(hfg : ∀ (x : α), x ∈ s → f x ≤ g x) :
theorem
measure_theory.lintegral_of_real_le_lintegral_nnnorm
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ℝ) :
theorem
measure_theory.lintegral_nnnorm_eq_of_ae_nonneg
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ℝ}
(h_nonneg : 0 ≤ᵐ[μ] f) :
theorem
measure_theory.lintegral_nnnorm_eq_of_nonneg
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ℝ}
(h_nonneg : 0 ≤ f) :
theorem
measure_theory.lintegral_supr
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(hf : ∀ (n : ℕ), measurable (f n))
(h_mono : monotone f) :
Monotone convergence theorem -- sometimes called Beppo-Levi convergence.
See lintegral_supr_directed for a more general form.
theorem
measure_theory.lintegral_supr'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(hf : ∀ (n : ℕ), ae_measurable (f n) μ)
(h_mono : ∀ᵐ (x : α) ∂μ, monotone (λ (n : ℕ), f n x)) :
Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with ae_measurable functions.
theorem
measure_theory.lintegral_tendsto_of_tendsto_of_monotone
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
{F : α → ennreal}
(hf : ∀ (n : ℕ), ae_measurable (f n) μ)
(h_mono : ∀ᵐ (x : α) ∂μ, monotone (λ (n : ℕ), f n x))
(h_tendsto : ∀ᵐ (x : α) ∂μ, filter.tendsto (λ (n : ℕ), f n x) filter.at_top (nhds (F x))) :
filter.tendsto (λ (n : ℕ), ∫⁻ (x : α), f n x ∂μ) filter.at_top (nhds (∫⁻ (x : α), F x ∂μ))
Monotone convergence theorem expressed with limits
theorem
measure_theory.lintegral_eq_supr_eapprox_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.exists_pos_set_lintegral_lt_of_measure_lt
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(h : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
{ε : ennreal}
(hε : ε ≠ 0) :
If f has finite integral, then ∫⁻ x in s, f x ∂μ is absolutely continuous in s: it tends
to zero as μ s tends to zero. This lemma states states this fact in terms of ε and δ.
theorem
measure_theory.tendsto_set_lintegral_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{ι : Type u_2}
{f : α → ennreal}
(h : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
{l : filter ι}
{s : ι → set α}
(hl : filter.tendsto (⇑μ ∘ s) l (nhds 0)) :
If f has finite integral, then ∫⁻ x in s, f x ∂μ is absolutely continuous in s: it tends
to zero as μ s tends to zero.
theorem
measure_theory.le_lintegral_add
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f g : α → ennreal) :
The sum of the lower Lebesgue integrals of two functions is less than or equal to the integral of their sum. The other inequality needs one of these functions to be (a.e.-)measurable.
theorem
measure_theory.lintegral_add_aux
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g) :
@[simp]
theorem
measure_theory.lintegral_add_left
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(g : α → ennreal) :
If f g : α → ℝ≥0∞ are two functions and one of them is (a.e.) measurable, then the Lebesgue
integral of f + g equals the sum of integrals. This lemma assumes that f is integrable, see also
measure_theory.lintegral_add_right and primed versions of these lemmas.
@[simp]
theorem
measure_theory.lintegral_add_right
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal)
{g : α → ennreal}
(hg : measurable g) :
If f g : α → ℝ≥0∞ are two functions and one of them is (a.e.) measurable, then the Lebesgue
integral of f + g equals the sum of integrals. This lemma assumes that g is integrable, see also
measure_theory.lintegral_add_left and primed versions of these lemmas.
@[simp]
theorem
measure_theory.lintegral_sum_measure
{α : Type u_1}
{m : measurable_space α}
{ι : Type u_2}
(f : α → ennreal)
(μ : ι → measure_theory.measure α) :
theorem
measure_theory.has_sum_lintegral_measure
{α : Type u_1}
{ι : Type u_2}
{m : measurable_space α}
(f : α → ennreal)
(μ : ι → measure_theory.measure α) :
@[simp]
theorem
measure_theory.lintegral_zero_measure
{α : Type u_1}
{m : measurable_space α}
(f : α → ennreal) :
theorem
measure_theory.set_lintegral_empty
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal) :
theorem
measure_theory.set_lintegral_measure_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : set α)
(f : α → ennreal)
(hs' : ⇑μ s = 0) :
@[simp]
theorem
measure_theory.lintegral_const_mul
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_const_mul''
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
{f : α → ennreal}
(hf : ae_measurable f μ) :
theorem
measure_theory.lintegral_mul_const
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_mul_const''
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
{f : α → ennreal}
(hf : ae_measurable f μ) :
theorem
measure_theory.lintegral_lintegral_mul
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{β : Type u_2}
[measurable_space β]
{ν : measure_theory.measure β}
{f : α → ennreal}
{g : β → ennreal}
(hf : ae_measurable f μ)
(hg : ae_measurable g ν) :
theorem
measure_theory.lintegral_rw₂
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f₁ f₁' : α → β}
{f₂ f₂' : α → γ}
(h₁ : f₁ =ᵐ[μ] f₁')
(h₂ : f₂ =ᵐ[μ] f₂')
(g : β → γ → ennreal) :
@[simp]
theorem
measure_theory.lintegral_indicator
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.lintegral_indicator₀
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal)
{s : set α}
(hs : measure_theory.null_measurable_set s μ) :
theorem
measure_theory.lintegral_indicator_const
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
(c : ennreal) :
theorem
measure_theory.set_lintegral_eq_const
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(r : ennreal) :
@[simp]
theorem
measure_theory.lintegral_indicator_one
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.lintegral_add_mul_meas_add_le_le_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hle : f ≤ᵐ[μ] g)
(hg : ae_measurable g μ)
(ε : ennreal) :
A version of Markov's inequality for two functions. It doesn't follow from the standard
Markov's inequality because we only assume measurability of g, not f.
theorem
measure_theory.mul_meas_ge_le_lintegral₀
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(ε : ennreal) :
Markov's inequality also known as Chebyshev's first inequality.
theorem
measure_theory.mul_meas_ge_le_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(ε : ennreal) :
Markov's inequality also known as Chebyshev's first inequality. For a version assuming
ae_measurable, see mul_meas_ge_le_lintegral₀.
theorem
measure_theory.lintegral_eq_top_of_measure_eq_top_ne_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(hμf : ⇑μ {x : α | f x = ⊤} ≠ 0) :
theorem
measure_theory.set_lintegral_eq_top_of_measure_eq_top_ne_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
{s : set α}
(hf : ae_measurable f (μ.restrict s))
(hμf : ⇑μ {x ∈ s | f x = ⊤} ≠ 0) :
theorem
measure_theory.measure_eq_top_of_lintegral_ne_top
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(hμf : ∫⁻ (x : α), f x ∂μ ≠ ⊤) :
theorem
measure_theory.measure_eq_top_of_set_lintegral_ne_top
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
{s : set α}
(hf : ae_measurable f (μ.restrict s))
(hμf : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤) :
theorem
measure_theory.meas_ge_le_lintegral_div
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
{ε : ennreal}
(hε : ε ≠ 0)
(hε' : ε ≠ ⊤) :
Markov's inequality also known as Chebyshev's first inequality.
@[simp]
theorem
measure_theory.lintegral_eq_zero_iff'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ) :
@[simp]
theorem
measure_theory.lintegral_eq_zero_iff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_pos_iff_support
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_supr_ae
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(hf : ∀ (n : ℕ), measurable (f n))
(h_mono : ∀ (n : ℕ), ∀ᵐ (a : α) ∂μ, f n a ≤ f n.succ a) :
Weaker version of the monotone convergence theorem
theorem
measure_theory.lintegral_sub_le'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f g : α → ennreal)
(hf : ae_measurable f μ) :
theorem
measure_theory.lintegral_sub_le
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f g : α → ennreal)
(hf : measurable f) :
theorem
measure_theory.lintegral_strict_mono_of_ae_le_of_frequently_ae_lt
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hg : ae_measurable g μ)
(hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
(h_le : f ≤ᵐ[μ] g)
(h : ∃ᵐ (x : α) ∂μ, f x ≠ g x) :
theorem
measure_theory.lintegral_strict_mono_of_ae_le_of_ae_lt_on
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hg : ae_measurable g μ)
(hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
(h_le : f ≤ᵐ[μ] g)
{s : set α}
(hμs : ⇑μ s ≠ 0)
(h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x < g x) :
theorem
measure_theory.set_lintegral_strict_mono
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
{s : set α}
(hsm : measurable_set s)
(hs : ⇑μ s ≠ 0)
(hg : measurable g)
(hfi : ∫⁻ (x : α) in s, f x ∂μ ≠ ⊤)
(h : ∀ᵐ (x : α) ∂μ, x ∈ s → f x < g x) :
theorem
measure_theory.lintegral_infi_ae
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h_meas : ∀ (n : ℕ), measurable (f n))
(h_mono : ∀ (n : ℕ), f n.succ ≤ᵐ[μ] f n)
(h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤) :
Monotone convergence theorem for nonincreasing sequences of functions
theorem
measure_theory.lintegral_infi
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h_meas : ∀ (n : ℕ), measurable (f n))
(h_anti : antitone f)
(h_fin : ∫⁻ (a : α), f 0 a ∂μ ≠ ⊤) :
Monotone convergence theorem for nonincreasing sequences of functions
theorem
measure_theory.lintegral_liminf_le'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h_meas : ∀ (n : ℕ), ae_measurable (f n) μ) :
∫⁻ (a : α), filter.liminf (λ (n : ℕ), f n a) filter.at_top ∂μ ≤ filter.liminf (λ (n : ℕ), ∫⁻ (a : α), f n a ∂μ) filter.at_top
Known as Fatou's lemma, version with ae_measurable functions
theorem
measure_theory.lintegral_liminf_le
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h_meas : ∀ (n : ℕ), measurable (f n)) :
∫⁻ (a : α), filter.liminf (λ (n : ℕ), f n a) filter.at_top ∂μ ≤ filter.liminf (λ (n : ℕ), ∫⁻ (a : α), f n a ∂μ) filter.at_top
Known as Fatou's lemma
theorem
measure_theory.limsup_lintegral_le
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
{g : α → ennreal}
(hf_meas : ∀ (n : ℕ), measurable (f n))
(h_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g)
(h_fin : ∫⁻ (a : α), g a ∂μ ≠ ⊤) :
filter.limsup (λ (n : ℕ), ∫⁻ (a : α), f n a ∂μ) filter.at_top ≤ ∫⁻ (a : α), filter.limsup (λ (n : ℕ), f n a) filter.at_top ∂μ
theorem
measure_theory.tendsto_lintegral_of_dominated_convergence
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{F : ℕ → α → ennreal}
{f : α → ennreal}
(bound : α → ennreal)
(hF_meas : ∀ (n : ℕ), measurable (F n))
(h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound)
(h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤)
(h_lim : ∀ᵐ (a : α) ∂μ, filter.tendsto (λ (n : ℕ), F n a) filter.at_top (nhds (f a))) :
filter.tendsto (λ (n : ℕ), ∫⁻ (a : α), F n a ∂μ) filter.at_top (nhds (∫⁻ (a : α), f a ∂μ))
Dominated convergence theorem for nonnegative functions
theorem
measure_theory.tendsto_lintegral_of_dominated_convergence'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{F : ℕ → α → ennreal}
{f : α → ennreal}
(bound : α → ennreal)
(hF_meas : ∀ (n : ℕ), ae_measurable (F n) μ)
(h_bound : ∀ (n : ℕ), F n ≤ᵐ[μ] bound)
(h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤)
(h_lim : ∀ᵐ (a : α) ∂μ, filter.tendsto (λ (n : ℕ), F n a) filter.at_top (nhds (f a))) :
filter.tendsto (λ (n : ℕ), ∫⁻ (a : α), F n a ∂μ) filter.at_top (nhds (∫⁻ (a : α), f a ∂μ))
Dominated convergence theorem for nonnegative functions which are just almost everywhere measurable.
theorem
measure_theory.tendsto_lintegral_filter_of_dominated_convergence
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{ι : Type u_2}
{l : filter ι}
[l.is_countably_generated]
{F : ι → α → ennreal}
{f : α → ennreal}
(bound : α → ennreal)
(hF_meas : ∀ᶠ (n : ι) in l, measurable (F n))
(h_bound : ∀ᶠ (n : ι) in l, ∀ᵐ (a : α) ∂μ, F n a ≤ bound a)
(h_fin : ∫⁻ (a : α), bound a ∂μ ≠ ⊤)
(h_lim : ∀ᵐ (a : α) ∂μ, filter.tendsto (λ (n : ι), F n a) l (nhds (f a))) :
Dominated convergence theorem for filters with a countable basis
theorem
measure_theory.lintegral_supr_directed_of_measurable
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable β]
{f : β → α → ennreal}
(hf : ∀ (b : β), measurable (f b))
(h_directed : directed has_le.le f) :
Monotone convergence for a supremum over a directed family and indexed by a countable type
theorem
measure_theory.lintegral_supr_directed
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable β]
{f : β → α → ennreal}
(hf : ∀ (b : β), ae_measurable (f b) μ)
(h_directed : directed has_le.le f) :
Monotone convergence for a supremum over a directed family and indexed by a countable type.
theorem
measure_theory.lintegral_Union₀
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable β]
{s : β → set α}
(hm : ∀ (i : β), measure_theory.null_measurable_set (s i) μ)
(hd : pairwise (measure_theory.ae_disjoint μ on s))
(f : α → ennreal) :
theorem
measure_theory.lintegral_Union
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable β]
{s : β → set α}
(hm : ∀ (i : β), measurable_set (s i))
(hd : pairwise (disjoint on s))
(f : α → ennreal) :
theorem
measure_theory.lintegral_bUnion₀
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{t : set β}
{s : β → set α}
(ht : t.countable)
(hm : ∀ (i : β), i ∈ t → measure_theory.null_measurable_set (s i) μ)
(hd : t.pairwise (measure_theory.ae_disjoint μ on s))
(f : α → ennreal) :
theorem
measure_theory.lintegral_bUnion
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{t : set β}
{s : β → set α}
(ht : t.countable)
(hm : ∀ (i : β), i ∈ t → measurable_set (s i))
(hd : t.pairwise_disjoint s)
(f : α → ennreal) :
theorem
measure_theory.lintegral_bUnion_finset₀
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : finset β}
{t : β → set α}
(hd : ↑s.pairwise (measure_theory.ae_disjoint μ on t))
(hm : ∀ (b : β), b ∈ s → measure_theory.null_measurable_set (t b) μ)
(f : α → ennreal) :
theorem
measure_theory.lintegral_bUnion_finset
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : finset β}
{t : β → set α}
(hd : ↑s.pairwise_disjoint t)
(hm : ∀ (b : β), b ∈ s → measurable_set (t b))
(f : α → ennreal) :
theorem
measure_theory.lintegral_max
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g) :
theorem
measure_theory.set_lintegral_max
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g)
(s : set α) :
theorem
measure_theory.lintegral_map
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mβ : measurable_space β}
{f : β → ennreal}
{g : α → β}
(hf : measurable f)
(hg : measurable g) :
theorem
measure_theory.lintegral_map'
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mβ : measurable_space β}
{f : β → ennreal}
{g : α → β}
(hf : ae_measurable f (measure_theory.measure.map g μ))
(hg : ae_measurable g μ) :
theorem
measure_theory.lintegral_map_le
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mβ : measurable_space β}
(f : β → ennreal)
{g : α → β}
(hg : measurable g) :
theorem
measure_theory.lintegral_comp
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_space β]
{f : β → ennreal}
{g : α → β}
(hf : measurable f)
(hg : measurable g) :
measure_theory.lintegral μ (f ∘ g) = ∫⁻ (a : β), f a ∂measure_theory.measure.map g μ
theorem
measure_theory.set_lintegral_map
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_space β]
{f : β → ennreal}
{g : α → β}
{s : set β}
(hs : measurable_set s)
(hf : measurable f)
(hg : measurable g) :
theorem
measure_theory.lintegral_indicator_const_comp
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mβ : measurable_space β}
{f : α → β}
{s : set β}
(hf : measurable f)
(hs : measurable_set s)
(c : ennreal) :
theorem
measurable_embedding.lintegral_map
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_space β]
{g : α → β}
(hg : measurable_embedding g)
(f : β → ennreal) :
If g : α → β is a measurable embedding and f : β → ℝ≥0∞ is any function (not necessarily
measurable), then ∫⁻ a, f a ∂(map g μ) = ∫⁻ a, f (g a) ∂μ. Compare with lintegral_map wich
applies to any measurable g : α → β but requires that f is measurable as well.
theorem
measure_theory.lintegral_map_equiv
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_space β]
(f : β → ennreal)
(g : α ≃ᵐ β) :
The lintegral transforms appropriately under a measurable equivalence g : α ≃ᵐ β.
(Compare lintegral_map, which applies to a wider class of functions g : α → β, but requires
measurability of the function being integrated.)
theorem
measure_theory.measure_preserving.lintegral_comp
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mb : measurable_space β}
{ν : measure_theory.measure β}
{g : α → β}
(hg : measure_theory.measure_preserving g μ ν)
{f : β → ennreal}
(hf : measurable f) :
theorem
measure_theory.measure_preserving.lintegral_comp_emb
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mb : measurable_space β}
{ν : measure_theory.measure β}
{g : α → β}
(hg : measure_theory.measure_preserving g μ ν)
(hge : measurable_embedding g)
(f : β → ennreal) :
theorem
measure_theory.measure_preserving.set_lintegral_comp_preimage
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mb : measurable_space β}
{ν : measure_theory.measure β}
{g : α → β}
(hg : measure_theory.measure_preserving g μ ν)
{s : set β}
(hs : measurable_set s)
{f : β → ennreal}
(hf : measurable f) :
theorem
measure_theory.measure_preserving.set_lintegral_comp_preimage_emb
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mb : measurable_space β}
{ν : measure_theory.measure β}
{g : α → β}
(hg : measure_theory.measure_preserving g μ ν)
(hge : measurable_embedding g)
(f : β → ennreal)
(s : set β) :
theorem
measure_theory.measure_preserving.set_lintegral_comp_emb
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
{mb : measurable_space β}
{ν : measure_theory.measure β}
{g : α → β}
(hg : measure_theory.measure_preserving g μ ν)
(hge : measurable_embedding g)
(f : β → ennreal)
(s : set α) :
@[protected, instance]
theorem
measure_theory.lintegral_dirac'
{α : Type u_1}
[measurable_space α]
(a : α)
{f : α → ennreal}
(hf : measurable f) :
∫⁻ (a : α), f a ∂measure_theory.measure.dirac a = f a
theorem
measure_theory.lintegral_dirac
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(a : α)
(f : α → ennreal) :
∫⁻ (a : α), f a ∂measure_theory.measure.dirac a = f a
theorem
measure_theory.set_lintegral_dirac'
{α : Type u_1}
[measurable_space α]
{a : α}
{f : α → ennreal}
(hf : measurable f)
{s : set α}
(hs : measurable_set s)
[decidable (a ∈ s)] :
theorem
measure_theory.set_lintegral_dirac
{α : Type u_1}
[measurable_space α]
{a : α}
(f : α → ennreal)
(s : set α)
[measurable_singleton_class α]
[decidable (a ∈ s)] :
theorem
measure_theory.lintegral_count'
{α : Type u_1}
[measurable_space α]
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_count
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(f : α → ennreal) :
theorem
ennreal.tsum_const_eq
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
(c : ennreal) :
theorem
ennreal.count_const_le_le_of_tsum_le
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
{a : α → ennreal}
(a_mble : measurable a)
{c : ennreal}
(tsum_le_c : ∑' (i : α), a i ≤ c)
{ε : ennreal}
(ε_ne_zero : ε ≠ 0)
(ε_ne_top : ε ≠ ⊤) :
⇑measure_theory.measure.count {i : α | ε ≤ a i} ≤ c / ε
Markov's inequality for the counting measure with hypothesis using tsum in ℝ≥0∞.
theorem
nnreal.count_const_le_le_of_tsum_le
{α : Type u_1}
[measurable_space α]
[measurable_singleton_class α]
{a : α → nnreal}
(a_mble : measurable a)
(a_summable : summable a)
{c : nnreal}
(tsum_le_c : ∑' (i : α), a i ≤ c)
{ε : nnreal}
(ε_ne_zero : ε ≠ 0) :
Markov's inequality for counting measure with hypothesis using tsum in ℝ≥0.
Lebesgue integral over finite and countable types and sets #
theorem
measure_theory.lintegral_countable'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[countable α]
[measurable_singleton_class α]
(f : α → ennreal) :
theorem
measure_theory.lintegral_singleton'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(a : α) :
theorem
measure_theory.lintegral_singleton
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_singleton_class α]
(f : α → ennreal)
(a : α) :
theorem
measure_theory.lintegral_insert
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_singleton_class α]
{a : α}
{s : set α}
(h : a ∉ s)
(f : α → ennreal) :
theorem
measure_theory.lintegral_finset
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_singleton_class α]
(s : finset α)
(f : α → ennreal) :
theorem
measure_theory.lintegral_fintype
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measurable_singleton_class α]
[fintype α]
(f : α → ennreal) :
theorem
measure_theory.lintegral_unique
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[unique α]
(f : α → ennreal) :
theorem
measure_theory.ae_lt_top
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤) :
theorem
measure_theory.ae_lt_top'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(h2f : ∫⁻ (x : α), f x ∂μ ≠ ⊤) :
theorem
measure_theory.set_lintegral_lt_top_of_is_compact
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[topological_space α]
[opens_measurable_space α]
{s : set α}
(hs : ⇑μ s ≠ ⊤)
(hsc : is_compact s)
{f : α → nnreal}
(hf : continuous f) :
theorem
is_finite_measure.lintegral_lt_top_of_bounded_to_ennreal
{α : Type u_1}
[measurable_space α]
(μ : measure_theory.measure α)
[μ_fin : measure_theory.is_finite_measure μ]
{f : α → ennreal}
(f_bdd : ∃ (c : nnreal), ∀ (x : α), f x ≤ ↑c) :
noncomputable
def
measure_theory.measure.with_density
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
(f : α → ennreal) :
Given a measure μ : measure α and a function f : α → ℝ≥0∞, μ.with_density f is the
measure such that for a measurable set s we have μ.with_density f s = ∫⁻ a in s, f a ∂μ.
Equations
- μ.with_density f = measure_theory.measure.of_measurable (λ (s : set α) (hs : measurable_set s), ∫⁻ (a : α) in s, f a ∂μ) _ _
Instances for measure_theory.measure.with_density
@[simp]
theorem
measure_theory.with_density_apply
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.with_density_congr_ae
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(h : f =ᵐ[μ] g) :
μ.with_density f = μ.with_density g
theorem
measure_theory.with_density_add_left
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(g : α → ennreal) :
μ.with_density (f + g) = μ.with_density f + μ.with_density g
theorem
measure_theory.with_density_add_right
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : α → ennreal)
{g : α → ennreal}
(hg : measurable g) :
μ.with_density (f + g) = μ.with_density f + μ.with_density g
theorem
measure_theory.with_density_add_measure
{α : Type u_1}
{m : measurable_space α}
(μ ν : measure_theory.measure α)
(f : α → ennreal) :
(μ + ν).with_density f = μ.with_density f + ν.with_density f
theorem
measure_theory.with_density_sum
{α : Type u_1}
{ι : Type u_2}
{m : measurable_space α}
(μ : ι → measure_theory.measure α)
(f : α → ennreal) :
(measure_theory.measure.sum μ).with_density f = measure_theory.measure.sum (λ (n : ι), (μ n).with_density f)
theorem
measure_theory.with_density_smul
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
{f : α → ennreal}
(hf : measurable f) :
μ.with_density (r • f) = r • μ.with_density f
theorem
measure_theory.with_density_smul'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
(f : α → ennreal)
(hr : r ≠ ⊤) :
μ.with_density (r • f) = r • μ.with_density f
theorem
measure_theory.is_finite_measure_with_density
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ∫⁻ (a : α), f a ∂μ ≠ ⊤) :
theorem
measure_theory.with_density_absolutely_continuous
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
(f : α → ennreal) :
(μ.with_density f).absolutely_continuous μ
@[simp]
theorem
measure_theory.with_density_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
μ.with_density 0 = 0
@[simp]
theorem
measure_theory.with_density_one
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
μ.with_density 1 = μ
theorem
measure_theory.with_density_tsum
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h : ∀ (i : ℕ), measurable (f i)) :
μ.with_density (∑' (n : ℕ), f n) = measure_theory.measure.sum (λ (n : ℕ), μ.with_density (f n))
theorem
measure_theory.with_density_indicator
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
(f : α → ennreal) :
μ.with_density (s.indicator f) = (μ.restrict s).with_density f
theorem
measure_theory.with_density_indicator_one
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s) :
μ.with_density (s.indicator 1) = μ.restrict s
theorem
measure_theory.with_density_of_real_mutually_singular
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ℝ}
(hf : measurable f) :
(μ.with_density (λ (x : α), ennreal.of_real (f x))).mutually_singular (μ.with_density (λ (x : α), ennreal.of_real (-f x)))
theorem
measure_theory.restrict_with_density
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{s : set α}
(hs : measurable_set s)
(f : α → ennreal) :
(μ.with_density f).restrict s = (μ.restrict s).with_density f
theorem
measure_theory.with_density_eq_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(h : μ.with_density f = 0) :
theorem
measure_theory.with_density_apply_eq_zero
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
{s : set α}
(hf : measurable f) :
theorem
measure_theory.ae_with_density_iff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{p : α → Prop}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.ae_with_density_iff_ae_restrict
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{p : α → Prop}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.ae_measurable_with_density_ennreal_iff
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → nnreal}
(hf : measurable f)
{g : α → ennreal} :
ae_measurable g (μ.with_density (λ (x : α), ↑(f x))) ↔ ae_measurable (λ (x : α), ↑(f x) * g x) μ
theorem
measure_theory.lintegral_with_density_eq_lintegral_mul
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f : α → ennreal}
(h_mf : measurable f)
{g : α → ennreal} :
This is Exercise 1.2.1 from [Tao10]. It allows you to express integration of a measurable
function with respect to (μ.with_density f) as an integral with respect to μ, called the base
measure. μ is often the Lebesgue measure, and in this circumstance f is the probability density
function, and (μ.with_density f) represents any continuous random variable as a
probability measure, such as the uniform distribution between 0 and 1, the Gaussian distribution,
the exponential distribution, the Beta distribution, or the Cauchy distribution (see Section 2.4
of [wasserman2004]). Thus, this method shows how to one can calculate expectations, variances,
and other moments as a function of the probability density function.
theorem
measure_theory.set_lintegral_with_density_eq_set_lintegral_mul
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.lintegral_with_density_eq_lintegral_mul₀'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
{g : α → ennreal}
(hg : ae_measurable g (μ.with_density f)) :
The Lebesgue integral of g with respect to the measure μ.with_density f coincides with
the integral of f * g. This version assumes that g is almost everywhere measurable. For a
version without conditions on g but requiring that f is almost everywhere finite, see
lintegral_with_density_eq_lintegral_mul_non_measurable
theorem
measure_theory.lintegral_with_density_eq_lintegral_mul₀
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
{g : α → ennreal}
(hg : ae_measurable g μ) :
theorem
measure_theory.lintegral_with_density_le_lintegral_mul
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f : α → ennreal}
(f_meas : measurable f)
(g : α → ennreal) :
theorem
measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f : α → ennreal}
(f_meas : measurable f)
(hf : ∀ᵐ (x : α) ∂μ, f x < ⊤)
(g : α → ennreal) :
theorem
measure_theory.set_lintegral_with_density_eq_set_lintegral_mul_non_measurable
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f : α → ennreal}
(f_meas : measurable f)
(g : α → ennreal)
{s : set α}
(hs : measurable_set s)
(hf : ∀ᵐ (x : α) ∂μ.restrict s, f x < ⊤) :
theorem
measure_theory.lintegral_with_density_eq_lintegral_mul_non_measurable₀
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f : α → ennreal}
(hf : ae_measurable f μ)
(h'f : ∀ᵐ (x : α) ∂μ, f x < ⊤)
(g : α → ennreal) :
theorem
measure_theory.set_lintegral_with_density_eq_set_lintegral_mul_non_measurable₀
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f : α → ennreal}
{s : set α}
(hf : ae_measurable f (μ.restrict s))
(g : α → ennreal)
(hs : measurable_set s)
(h'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ⊤) :
theorem
measure_theory.with_density_mul
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g) :
μ.with_density (f * g) = (μ.with_density f).with_density g
theorem
measure_theory.exists_pos_lintegral_lt_of_sigma_finite
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ]
{ε : ennreal}
(ε0 : ε ≠ 0) :
In a sigma-finite measure space, there exists an integrable function which is positive everywhere (and with an arbitrarily small integral).
theorem
measure_theory.lintegral_trim
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_trim_ae
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
{f : α → ennreal}
(hf : ae_measurable f (μ.trim hm)) :
theorem
measure_theory.univ_le_of_forall_fin_meas_le
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
(C : ennreal)
{f : set α → ennreal}
(hf : ∀ (s : set α), measurable_set s → ⇑μ s ≠ ⊤ → f s ≤ C)
(h_F_lim : ∀ (S : ℕ → set α), (∀ (n : ℕ), measurable_set (S n)) → monotone S → (f (⋃ (n : ℕ), S n) ≤ ⨆ (n : ℕ), f (S n))) :
theorem
measure_theory.lintegral_le_of_forall_fin_meas_le_of_measurable
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
(C : ennreal)
{f : α → ennreal}
(hf_meas : measurable f)
(hf : ∀ (s : set α), measurable_set s → ⇑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) :
If the Lebesgue integral of a function is bounded by some constant on all sets with finite
measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral
over the whole space is bounded by that same constant. Version for a measurable function.
See lintegral_le_of_forall_fin_meas_le' for the more general ae_measurable version.
theorem
measure_theory.lintegral_le_of_forall_fin_meas_le'
{α : Type u_1}
{m m0 : measurable_space α}
{μ : measure_theory.measure α}
(hm : m ≤ m0)
[measure_theory.sigma_finite (μ.trim hm)]
(C : ennreal)
{f : α → ennreal}
(hf_meas : ae_measurable f μ)
(hf : ∀ (s : set α), measurable_set s → ⇑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) :
If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure in a sub-σ-algebra and the measure is σ-finite on that sub-σ-algebra, then the integral over the whole space is bounded by that same constant.
theorem
measure_theory.lintegral_le_of_forall_fin_meas_le
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
(C : ennreal)
{f : α → ennreal}
(hf_meas : ae_measurable f μ)
(hf : ∀ (s : set α), measurable_set s → ⇑μ s ≠ ⊤ → ∫⁻ (x : α) in s, f x ∂μ ≤ C) :
If the Lebesgue integral of a function is bounded by some constant on all sets with finite measure and the measure is σ-finite, then the integral over the whole space is bounded by that same constant.
theorem
measure_theory.simple_func.exists_lt_lintegral_simple_func_of_lt_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
{f : measure_theory.simple_func α nnreal}
{L : ennreal}
(hL : L < ∫⁻ (x : α), ↑(⇑f x) ∂μ) :
theorem
measure_theory.exists_lt_lintegral_simple_func_of_lt_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
[measure_theory.sigma_finite μ]
{f : α → nnreal}
{L : ennreal}
(hL : L < ∫⁻ (x : α), ↑(f x) ∂μ) :
theorem
measure_theory.exists_absolutely_continuous_is_finite_measure
{α : Type u_1}
{m : measurable_space α}
(μ : measure_theory.measure α)
[measure_theory.sigma_finite μ] :
A sigma-finite measure is absolutely continuous with respect to some finite measure.