Lebesgue integral for ennreal-valued functions
We define simple functions and show that each Borel measurable function on ennreal can be
approximated by a sequence of simple functions.
To prove something for an arbitrary measurable function into ennreal, the theorem
measurable.ennreal_induction shows that is it sufficient to show that the property holds for
(multiples of) characteristic functions and is closed under addition and supremum of increasing
sequences of functions.
Notation
We introduce the following notation for the lower Lebesgue integral of a function f : α → ennreal.
∫⁻ x, f x ∂μ: integral of a functionf : α → ennrealwith respect to a measureμ;∫⁻ x, f x: integral of a functionf : α → ennrealwith respect to the canonical measurevolumeonα;∫⁻ x in s, f x ∂μ: integral of a functionf : α → ennrealover a setswith respect to a measureμ, defined as∫⁻ x, f x ∂(μ.restrict s);∫⁻ x in s, f x: integral of a functionf : α → ennrealover a setswith respect to the canonical measurevolume, defined as∫⁻ x, f x ∂(volume.restrict s).
structure
measure_theory.simple_func
(α : Type u)
[measurable_space α]
(β : Type v) :
Type (max u v)
- to_fun : α → β
- is_measurable_fiber' : ∀ (x : β), is_measurable (c.to_fun ⁻¹' {x})
- finite_range' : (set.range c.to_fun).finite
A function f from a measurable space to any type is called simple,
if every preimage f ⁻¹' {x} is measurable, and the range is finite. This structure bundles
a function with these properties.
@[instance]
Equations
- measure_theory.simple_func.has_coe_to_fun = {F := λ (x : measure_theory.simple_func α β), α → β, coe := measure_theory.simple_func.to_fun β}
theorem
measure_theory.simple_func.coe_injective
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
⦃f g : measure_theory.simple_func α β⦄
(H : ⇑f = ⇑g) :
f = g
@[ext]
theorem
measure_theory.simple_func.ext
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{f g : measure_theory.simple_func α β}
(H : ∀ (a : α), ⇑f a = ⇑g a) :
f = g
theorem
measure_theory.simple_func.finite_range
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.is_measurable_fiber
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(x : β) :
is_measurable (⇑f ⁻¹' {x})
def
measure_theory.simple_func.range
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β) :
finset β
Range of a simple function α →ₛ β as a finset β.
@[simp]
theorem
measure_theory.simple_func.mem_range
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{f : measure_theory.simple_func α β}
{b : β} :
theorem
measure_theory.simple_func.mem_range_self
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(x : α) :
@[simp]
theorem
measure_theory.simple_func.coe_range
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.mem_range_of_measure_ne_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{f : measure_theory.simple_func α β}
{x : β}
{μ : measure_theory.measure α}
(H : ⇑μ (⇑f ⁻¹' {x}) ≠ 0) :
theorem
measure_theory.simple_func.forall_range_iff
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{f : measure_theory.simple_func α β}
{p : β → Prop} :
theorem
measure_theory.simple_func.exists_range_iff
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{f : measure_theory.simple_func α β}
{p : β → Prop} :
theorem
measure_theory.simple_func.preimage_eq_empty_iff
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(b : β) :
Constant function as a simple_func.
Equations
- measure_theory.simple_func.const α b = {to_fun := λ (a : α), b, is_measurable_fiber' := _, finite_range' := _}
@[instance]
def
measure_theory.simple_func.inhabited
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[inhabited β] :
Equations
theorem
measure_theory.simple_func.const_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(a : α)
(b : β) :
⇑(measure_theory.simple_func.const α b) a = b
@[simp]
theorem
measure_theory.simple_func.coe_const
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(b : β) :
⇑(measure_theory.simple_func.const α b) = function.const α b
@[simp]
theorem
measure_theory.simple_func.range_const
{β : Type u_2}
(α : Type u_1)
[measurable_space α]
[nonempty α]
(b : β) :
(measure_theory.simple_func.const α b).range = {b}
theorem
measure_theory.simple_func.is_measurable_cut
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(r : α → β → Prop)
(f : measure_theory.simple_func α β)
(h : ∀ (b : β), is_measurable {a : α | r a b}) :
is_measurable {a : α | r a (⇑f a)}
theorem
measure_theory.simple_func.is_measurable_preimage
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(s : set β) :
is_measurable (⇑f ⁻¹' s)
theorem
measure_theory.simple_func.measurable
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
(f : measure_theory.simple_func α β) :
A simple function is measurable
theorem
measure_theory.simple_func.sum_measure_preimage_singleton
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
{μ : measure_theory.measure α}
(s : finset β) :
theorem
measure_theory.simple_func.sum_range_measure_preimage_singleton
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(μ : measure_theory.measure α) :
def
measure_theory.simple_func.piecewise
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(s : set α)
(hs : is_measurable s)
(f g : measure_theory.simple_func α β) :
If-then-else as a simple_func.
Equations
- measure_theory.simple_func.piecewise s hs f g = {to_fun := s.piecewise ⇑f ⇑g (λ (j : α), s.decidable_mem j), is_measurable_fiber' := _, finite_range' := _}
@[simp]
theorem
measure_theory.simple_func.coe_piecewise
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{s : set α}
(hs : is_measurable s)
(f g : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.piecewise_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{s : set α}
(hs : is_measurable s)
(f g : measure_theory.simple_func α β)
(a : α) :
@[simp]
theorem
measure_theory.simple_func.piecewise_compl
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{s : set α}
(hs : is_measurable sᶜ)
(f g : measure_theory.simple_func α β) :
measure_theory.simple_func.piecewise sᶜ hs f g = measure_theory.simple_func.piecewise s _ g f
@[simp]
theorem
measure_theory.simple_func.piecewise_univ
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f g : measure_theory.simple_func α β) :
@[simp]
theorem
measure_theory.simple_func.piecewise_empty
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f g : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.measurable_bind
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space γ]
(f : measure_theory.simple_func α β)
(g : β → α → γ)
(hg : ∀ (b : β), measurable (g b)) :
measurable (λ (a : α), g (⇑f a) a)
def
measure_theory.simple_func.bind
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : β → measure_theory.simple_func α γ) :
If f : α →ₛ β is a simple function and g : β → α →ₛ γ is a family of simple functions,
then f.bind g binds the first argument of g to f. In other words, f.bind g a = g (f a) a.
Equations
- f.bind g = {to_fun := λ (a : α), ⇑(g (⇑f a)) a, is_measurable_fiber' := _, finite_range' := _}
@[simp]
theorem
measure_theory.simple_func.bind_apply
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : β → measure_theory.simple_func α γ)
(a : α) :
def
measure_theory.simple_func.map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(g : β → γ)
(f : measure_theory.simple_func α β) :
Given a function g : β → γ and a simple function f : α →ₛ β, f.map g return the simple
function g ∘ f : α →ₛ γ
Equations
theorem
measure_theory.simple_func.map_apply
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(g : β → γ)
(f : measure_theory.simple_func α β)
(a : α) :
⇑(measure_theory.simple_func.map g f) a = g (⇑f a)
theorem
measure_theory.simple_func.map_map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{δ : Type u_4}
[measurable_space α]
(g : β → γ)
(h : γ → δ)
(f : measure_theory.simple_func α β) :
@[simp]
theorem
measure_theory.simple_func.coe_map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(g : β → γ)
(f : measure_theory.simple_func α β) :
⇑(measure_theory.simple_func.map g f) = g ∘ ⇑f
@[simp]
theorem
measure_theory.simple_func.range_map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[decidable_eq γ]
(g : β → γ)
(f : measure_theory.simple_func α β) :
(measure_theory.simple_func.map g f).range = finset.image g f.range
@[simp]
theorem
measure_theory.simple_func.map_const
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(g : β → γ)
(b : β) :
theorem
measure_theory.simple_func.map_preimage
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : β → γ)
(s : set γ) :
⇑(measure_theory.simple_func.map g f) ⁻¹' s = ⇑f ⁻¹' ↑(finset.filter (λ (b : β), g b ∈ s) f.range)
theorem
measure_theory.simple_func.map_preimage_singleton
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : β → γ)
(c : γ) :
⇑(measure_theory.simple_func.map g f) ⁻¹' {c} = ⇑f ⁻¹' ↑(finset.filter (λ (b : β), g b = c) f.range)
def
measure_theory.simple_func.comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f : measure_theory.simple_func β γ)
(g : α → β)
(hgm : measurable g) :
Composition of a simple_fun and a measurable function is a simple_func.
Equations
- f.comp g hgm = {to_fun := ⇑f ∘ g, is_measurable_fiber' := _, finite_range' := _}
@[simp]
theorem
measure_theory.simple_func.coe_comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f : measure_theory.simple_func β γ)
{g : α → β}
(hgm : measurable g) :
theorem
measure_theory.simple_func.range_comp_subset_range
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f : measure_theory.simple_func β γ)
{g : α → β}
(hgm : measurable g) :
def
measure_theory.simple_func.seq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α (β → γ))
(g : measure_theory.simple_func α β) :
If f is a simple function taking values in β → γ and g is another simple function
with the same domain and codomain β, then f.seq g = f a (g a).
Equations
- f.seq g = f.bind (λ (f : β → γ), measure_theory.simple_func.map f g)
@[simp]
theorem
measure_theory.simple_func.seq_apply
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α (β → γ))
(g : measure_theory.simple_func α β)
(a : α) :
def
measure_theory.simple_func.pair
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : measure_theory.simple_func α γ) :
measure_theory.simple_func α (β × γ)
Combine two simple functions f : α →ₛ β and g : α →ₛ β
into λ a, (f a, g a).
Equations
- f.pair g = (measure_theory.simple_func.map prod.mk f).seq g
@[simp]
theorem
measure_theory.simple_func.pair_apply
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : measure_theory.simple_func α γ)
(a : α) :
theorem
measure_theory.simple_func.pair_preimage
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : measure_theory.simple_func α γ)
(s : set β)
(t : set γ) :
theorem
measure_theory.simple_func.pair_preimage_singleton
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
(f : measure_theory.simple_func α β)
(g : measure_theory.simple_func α γ)
(b : β)
(c : γ) :
theorem
measure_theory.simple_func.bind_const
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β) :
f.bind (measure_theory.simple_func.const α) = f
@[instance]
def
measure_theory.simple_func.has_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β] :
Equations
@[instance]
def
measure_theory.simple_func.has_add
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_add β] :
Equations
- measure_theory.simple_func.has_add = {add := λ (f g : measure_theory.simple_func α β), (measure_theory.simple_func.map has_add.add f).seq g}
@[instance]
def
measure_theory.simple_func.has_mul
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_mul β] :
Equations
- measure_theory.simple_func.has_mul = {mul := λ (f g : measure_theory.simple_func α β), (measure_theory.simple_func.map has_mul.mul f).seq g}
@[instance]
def
measure_theory.simple_func.has_sup
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sup β] :
Equations
- measure_theory.simple_func.has_sup = {sup := λ (f g : measure_theory.simple_func α β), (measure_theory.simple_func.map has_sup.sup f).seq g}
@[instance]
def
measure_theory.simple_func.has_inf
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_inf β] :
Equations
- measure_theory.simple_func.has_inf = {inf := λ (f g : measure_theory.simple_func α β), (measure_theory.simple_func.map has_inf.inf f).seq g}
@[instance]
def
measure_theory.simple_func.has_le
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_le β] :
Equations
- measure_theory.simple_func.has_le = {le := λ (f g : measure_theory.simple_func α β), ∀ (a : α), ⇑f a ≤ ⇑g a}
@[simp]
theorem
measure_theory.simple_func.coe_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β] :
@[simp]
theorem
measure_theory.simple_func.const_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β] :
@[simp]
theorem
measure_theory.simple_func.coe_add
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_add β]
(f g : measure_theory.simple_func α β) :
@[simp]
theorem
measure_theory.simple_func.coe_mul
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_mul β]
(f g : measure_theory.simple_func α β) :
@[simp]
theorem
measure_theory.simple_func.coe_le
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[preorder β]
{f g : measure_theory.simple_func α β} :
@[simp]
theorem
measure_theory.simple_func.range_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[nonempty α]
[has_zero β] :
theorem
measure_theory.simple_func.eq_zero_of_mem_range_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{y : β}
(a : y ∈ 0.range) :
y = 0
theorem
measure_theory.simple_func.sup_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sup β]
(f g : measure_theory.simple_func α β)
(a : α) :
theorem
measure_theory.simple_func.mul_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_mul β]
(f g : measure_theory.simple_func α β)
(a : α) :
theorem
measure_theory.simple_func.add_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_add β]
(f g : measure_theory.simple_func α β)
(a : α) :
theorem
measure_theory.simple_func.add_eq_map₂
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_add β]
(f g : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.mul_eq_map₂
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_mul β]
(f g : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.sup_eq_map₂
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sup β]
(f g : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.const_mul_eq_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_mul β]
(f : measure_theory.simple_func α β)
(b : β) :
(measure_theory.simple_func.const α b) * f = measure_theory.simple_func.map (λ (a : β), b * a) f
@[instance]
def
measure_theory.simple_func.add_monoid
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[add_monoid β] :
Equations
- measure_theory.simple_func.add_monoid = function.injective.add_monoid (λ (f : measure_theory.simple_func α β), show α → β, from ⇑f) measure_theory.simple_func.coe_injective measure_theory.simple_func.add_monoid._proof_1 measure_theory.simple_func.add_monoid._proof_2
@[instance]
def
measure_theory.simple_func.add_comm_monoid
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[add_comm_monoid β] :
Equations
- measure_theory.simple_func.add_comm_monoid = function.injective.add_comm_monoid (λ (f : measure_theory.simple_func α β), show α → β, from ⇑f) measure_theory.simple_func.coe_injective measure_theory.simple_func.add_comm_monoid._proof_1 measure_theory.simple_func.add_comm_monoid._proof_2
@[instance]
def
measure_theory.simple_func.has_neg
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_neg β] :
Equations
@[simp]
theorem
measure_theory.simple_func.coe_neg
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_neg β]
(f : measure_theory.simple_func α β) :
@[instance]
def
measure_theory.simple_func.add_group
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[add_group β] :
Equations
- measure_theory.simple_func.add_group = function.injective.add_group (λ (f : measure_theory.simple_func α β), show α → β, from ⇑f) measure_theory.simple_func.coe_injective measure_theory.simple_func.add_group._proof_1 measure_theory.simple_func.add_group._proof_2 measure_theory.simple_func.add_group._proof_3
@[simp]
theorem
measure_theory.simple_func.coe_sub
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[add_group β]
(f g : measure_theory.simple_func α β) :
@[instance]
def
measure_theory.simple_func.add_comm_group
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[add_comm_group β] :
Equations
- measure_theory.simple_func.add_comm_group = function.injective.add_comm_group (λ (f : measure_theory.simple_func α β), show α → β, from ⇑f) measure_theory.simple_func.coe_injective measure_theory.simple_func.add_comm_group._proof_1 measure_theory.simple_func.add_comm_group._proof_2 measure_theory.simple_func.add_comm_group._proof_3
@[instance]
def
measure_theory.simple_func.has_scalar
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[has_scalar K β] :
has_scalar K (measure_theory.simple_func α β)
Equations
- measure_theory.simple_func.has_scalar = {smul := λ (k : K) (f : measure_theory.simple_func α β), measure_theory.simple_func.map (has_scalar.smul k) f}
@[simp]
theorem
measure_theory.simple_func.coe_smul
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[has_scalar K β]
(c : K)
(f : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.smul_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[has_scalar K β]
(k : K)
(f : measure_theory.simple_func α β)
(a : α) :
@[instance]
def
measure_theory.simple_func.semimodule
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[semiring K]
[add_comm_monoid β]
[semimodule K β] :
semimodule K (measure_theory.simple_func α β)
Equations
- measure_theory.simple_func.semimodule = function.injective.semimodule K {to_fun := λ (f : measure_theory.simple_func α β), show α → β, from ⇑f, map_zero' := _, map_add' := _} measure_theory.simple_func.coe_injective measure_theory.simple_func.semimodule._proof_3
theorem
measure_theory.simple_func.smul_eq_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[has_scalar K β]
(k : K)
(f : measure_theory.simple_func α β) :
k • f = measure_theory.simple_func.map (has_scalar.smul k) f
@[instance]
def
measure_theory.simple_func.preorder
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[preorder β] :
Equations
- measure_theory.simple_func.preorder = {le := has_le.le measure_theory.simple_func.has_le, lt := λ (a b : measure_theory.simple_func α β), a ≤ b ∧ ¬b ≤ a, le_refl := _, le_trans := _, lt_iff_le_not_le := _}
@[instance]
def
measure_theory.simple_func.partial_order
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[partial_order β] :
Equations
@[instance]
def
measure_theory.simple_func.order_bot
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[order_bot β] :
Equations
@[instance]
def
measure_theory.simple_func.order_top
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[order_top β] :
Equations
@[instance]
def
measure_theory.simple_func.semilattice_inf
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_inf β] :
Equations
- measure_theory.simple_func.semilattice_inf = {inf := has_inf.inf measure_theory.simple_func.has_inf, le := partial_order.le measure_theory.simple_func.partial_order, lt := partial_order.lt measure_theory.simple_func.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, inf_le_left := _, inf_le_right := _, le_inf := _}
@[instance]
def
measure_theory.simple_func.semilattice_sup
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_sup β] :
Equations
- measure_theory.simple_func.semilattice_sup = {sup := has_sup.sup measure_theory.simple_func.has_sup, le := partial_order.le measure_theory.simple_func.partial_order, lt := partial_order.lt measure_theory.simple_func.partial_order, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _}
@[instance]
def
measure_theory.simple_func.semilattice_sup_bot
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_sup_bot β] :
Equations
- measure_theory.simple_func.semilattice_sup_bot = {bot := order_bot.bot measure_theory.simple_func.order_bot, le := semilattice_sup.le measure_theory.simple_func.semilattice_sup, lt := semilattice_sup.lt measure_theory.simple_func.semilattice_sup, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, bot_le := _, sup := semilattice_sup.sup measure_theory.simple_func.semilattice_sup, le_sup_left := _, le_sup_right := _, sup_le := _}
@[instance]
def
measure_theory.simple_func.lattice
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[lattice β] :
Equations
- measure_theory.simple_func.lattice = {sup := semilattice_sup.sup measure_theory.simple_func.semilattice_sup, le := semilattice_sup.le measure_theory.simple_func.semilattice_sup, lt := semilattice_sup.lt measure_theory.simple_func.semilattice_sup, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := semilattice_inf.inf measure_theory.simple_func.semilattice_inf, inf_le_left := _, inf_le_right := _, le_inf := _}
@[instance]
def
measure_theory.simple_func.bounded_lattice
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[bounded_lattice β] :
Equations
- measure_theory.simple_func.bounded_lattice = {sup := lattice.sup measure_theory.simple_func.lattice, le := lattice.le measure_theory.simple_func.lattice, lt := lattice.lt measure_theory.simple_func.lattice, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := lattice.inf measure_theory.simple_func.lattice, inf_le_left := _, inf_le_right := _, le_inf := _, top := order_top.top measure_theory.simple_func.order_top, le_top := _, bot := order_bot.bot measure_theory.simple_func.order_bot, bot_le := _}
theorem
measure_theory.simple_func.finset_sup_apply
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[semilattice_sup_bot β]
{f : γ → measure_theory.simple_func α β}
(s : finset γ)
(a : α) :
def
measure_theory.simple_func.restrict
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β)
(s : set α) :
Restrict a simple function f : α →ₛ β to a set s. If s is measurable,
then f.restrict s a = if a ∈ s then f a else 0, otherwise f.restrict s = const α 0.
Equations
- f.restrict s = dite (is_measurable s) (λ (hs : is_measurable s), measure_theory.simple_func.piecewise s hs f 0) (λ (hs : ¬is_measurable s), 0)
theorem
measure_theory.simple_func.restrict_of_not_measurable
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{f : measure_theory.simple_func α β}
{s : set α}
(hs : ¬is_measurable s) :
@[simp]
theorem
measure_theory.simple_func.coe_restrict
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β)
{s : set α}
(hs : is_measurable s) :
@[simp]
theorem
measure_theory.simple_func.restrict_univ
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β) :
@[simp]
theorem
measure_theory.simple_func.restrict_empty
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.map_restrict_of_zero
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_zero β]
[has_zero γ]
{g : β → γ}
(hg : g 0 = 0)
(f : measure_theory.simple_func α β)
(s : set α) :
measure_theory.simple_func.map g (f.restrict s) = (measure_theory.simple_func.map g f).restrict s
theorem
measure_theory.simple_func.map_coe_ennreal_restrict
{α : Type u_1}
[measurable_space α]
(f : measure_theory.simple_func α ℝ≥0)
(s : set α) :
theorem
measure_theory.simple_func.map_coe_nnreal_restrict
{α : Type u_1}
[measurable_space α]
(f : measure_theory.simple_func α ℝ≥0)
(s : set α) :
theorem
measure_theory.simple_func.restrict_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β)
{s : set α}
(hs : is_measurable s)
(a : α) :
theorem
measure_theory.simple_func.restrict_preimage
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β)
{s : set α}
(hs : is_measurable s)
{t : set β}
(ht : 0 ∉ t) :
theorem
measure_theory.simple_func.restrict_preimage_singleton
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β)
{s : set α}
(hs : is_measurable s)
{r : β}
(hr : r ≠ 0) :
theorem
measure_theory.simple_func.mem_restrict_range
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{r : β}
{s : set α}
{f : measure_theory.simple_func α β}
(hs : is_measurable s) :
theorem
measure_theory.simple_func.mem_image_of_mem_range_restrict
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{r : β}
{s : set α}
{f : measure_theory.simple_func α β}
(hr : r ∈ (f.restrict s).range)
(h0 : r ≠ 0) :
theorem
measure_theory.simple_func.restrict_mono
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
[preorder β]
(s : set α)
{f g : measure_theory.simple_func α β}
(H : f ≤ g) :
def
measure_theory.simple_func.approx
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_sup_bot β]
[has_zero β]
(i : ℕ → β)
(f : α → β)
(n : ℕ) :
Fix a sequence i : ℕ → β. Given a function α → β, its n-th approximation
by simple functions is defined so that in case β = ennreal it sends each a to the supremum
of the set {i k | k ≤ n ∧ i k ≤ f a}, see approx_apply and supr_approx_apply for details.
Equations
- measure_theory.simple_func.approx i f n = (finset.range n).sup (λ (k : ℕ), (measure_theory.simple_func.const α (i k)).restrict {a : α | i k ≤ f a})
theorem
measure_theory.simple_func.approx_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_sup_bot β]
[has_zero β]
[topological_space β]
[order_closed_topology β]
[measurable_space β]
[opens_measurable_space β]
{i : ℕ → β}
{f : α → β}
{n : ℕ}
(a : α)
(hf : measurable f) :
⇑(measure_theory.simple_func.approx i f n) a = (finset.range n).sup (λ (k : ℕ), ite (i k ≤ f a) (i k) 0)
theorem
measure_theory.simple_func.monotone_approx
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_sup_bot β]
[has_zero β]
(i : ℕ → β)
(f : α → β) :
theorem
measure_theory.simple_func.approx_comp
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[semilattice_sup_bot β]
[has_zero β]
[topological_space β]
[order_closed_topology β]
[measurable_space β]
[opens_measurable_space β]
[measurable_space γ]
{i : ℕ → β}
{f : γ → β}
{g : α → γ}
{n : ℕ}
(a : α)
(hf : measurable f)
(hg : measurable g) :
⇑(measure_theory.simple_func.approx i (f ∘ g) n) a = ⇑(measure_theory.simple_func.approx i f n) (g a)
theorem
measure_theory.simple_func.supr_approx_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[topological_space β]
[complete_lattice β]
[order_closed_topology β]
[has_zero β]
[measurable_space β]
[opens_measurable_space β]
(i : ℕ → β)
(f : α → β)
(a : α)
(hf : measurable f)
(h_zero : 0 = ⊥) :
A sequence of ennreals such that its range is the set of non-negative rational numbers.
Equations
def
measure_theory.simple_func.eapprox
{α : Type u_1}
[measurable_space α]
(a : α → ennreal)
(a_1 : ℕ) :
Approximate a function α → ennreal by a sequence of simple functions.
theorem
measure_theory.simple_func.monotone_eapprox
{α : Type u_1}
[measurable_space α]
(f : α → ennreal) :
theorem
measure_theory.simple_func.supr_eapprox_apply
{α : Type u_1}
[measurable_space α]
(f : α → ennreal)
(hf : measurable f)
(a : α) :
(⨆ (n : ℕ), ⇑(measure_theory.simple_func.eapprox f n) a) = f a
theorem
measure_theory.simple_func.eapprox_comp
{α : Type u_1}
{γ : Type u_3}
[measurable_space α]
[measurable_space γ]
{f : γ → ennreal}
{g : α → γ}
{n : ℕ}
(hf : measurable f)
(hg : measurable g) :
⇑(measure_theory.simple_func.eapprox (f ∘ g) n) = ⇑(measure_theory.simple_func.eapprox f n) ∘ g
def
measure_theory.simple_func.lintegral
{α : Type u_1}
[measurable_space α]
(f : measure_theory.simple_func α ennreal)
(μ : measure_theory.measure α) :
Integral of a simple function whose codomain is ennreal.
theorem
measure_theory.simple_func.lintegral_eq_of_subset
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
{s : finset ennreal}
(hs : ∀ (x : α), ⇑f x ≠ 0 → ⇑μ (⇑f ⁻¹' {⇑f x}) ≠ 0 → ⇑f x ∈ s) :
theorem
measure_theory.simple_func.map_lintegral
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
(g : β → ennreal)
(f : measure_theory.simple_func α β) :
Calculate the integral of (g ∘ f), where g : β → ennreal and f : α →ₛ β.
theorem
measure_theory.simple_func.add_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f g : measure_theory.simple_func α ennreal) :
theorem
measure_theory.simple_func.const_mul_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
(x : ennreal) :
Integral of a simple function α →ₛ ennreal as a bilinear map.
@[simp]
theorem
measure_theory.simple_func.zero_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α} :
theorem
measure_theory.simple_func.lintegral_add
{α : Type u_1}
[measurable_space α]
{μ ν : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal) :
theorem
measure_theory.simple_func.lintegral_smul
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
(c : ennreal) :
@[simp]
theorem
measure_theory.simple_func.lintegral_zero
{α : Type u_1}
[measurable_space α]
(f : measure_theory.simple_func α ennreal) :
theorem
measure_theory.simple_func.lintegral_sum
{α : Type u_1}
[measurable_space α]
{ι : Type u_2}
(f : measure_theory.simple_func α ennreal)
(μ : ι → measure_theory.measure α) :
f.lintegral (measure_theory.measure.sum μ) = ∑' (i : ι), f.lintegral (μ i)
theorem
measure_theory.simple_func.restrict_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
{s : set α}
(hs : is_measurable s) :
theorem
measure_theory.simple_func.lintegral_restrict
{α : Type u_1}
[measurable_space α]
(f : measure_theory.simple_func α ennreal)
(s : set α)
(μ : measure_theory.measure α) :
theorem
measure_theory.simple_func.restrict_lintegral_eq_lintegral_restrict
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
{s : set α}
(hs : is_measurable s) :
theorem
measure_theory.simple_func.const_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(c : ennreal) :
theorem
measure_theory.simple_func.const_lintegral_restrict
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(c : ennreal)
(s : set α) :
theorem
measure_theory.simple_func.restrict_const_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(c : ennreal)
{s : set α}
(hs : is_measurable s) :
theorem
measure_theory.simple_func.le_sup_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f g : measure_theory.simple_func α ennreal) :
theorem
measure_theory.simple_func.lintegral_mono
{α : Type u_1}
[measurable_space α]
{f g : measure_theory.simple_func α ennreal}
(hfg : f ≤ g)
{μ ν : measure_theory.measure α}
(hμν : μ ≤ ν) :
simple_func.lintegral is monotone both in function and in measure.
theorem
measure_theory.simple_func.lintegral_eq_of_measure_preimage
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[measurable_space β]
{f : measure_theory.simple_func α ennreal}
{g : measure_theory.simple_func β ennreal}
{ν : measure_theory.measure β}
(H : ∀ (y : ennreal), ⇑μ (⇑f ⁻¹' {y}) = ⇑ν (⇑g ⁻¹' {y})) :
simple_func.lintegral depends only on the measures of f ⁻¹' {y}.
theorem
measure_theory.simple_func.lintegral_congr
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : measure_theory.simple_func α ennreal}
(h : ⇑f =ᵐ[μ] ⇑g) :
If two simple functions are equal a.e., then their lintegrals are equal.
theorem
measure_theory.simple_func.lintegral_map
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{β : Type u_2}
[measurable_space β]
{μ' : measure_theory.measure β}
(f : measure_theory.simple_func α ennreal)
(g : measure_theory.simple_func β ennreal)
(m : α → β)
(eq : ∀ (a : α), ⇑f a = ⇑g (m a))
(h : ∀ (s : set β), is_measurable s → ⇑μ' s = ⇑μ (m ⁻¹' s)) :
theorem
measure_theory.simple_func.support_eq
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β) :
function.support ⇑f = ⋃ (y : β) (H : y ∈ finset.filter (λ (y : β), y ≠ 0) f.range), ⇑f ⁻¹' {y}
def
measure_theory.simple_func.fin_meas_supp
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
(f : measure_theory.simple_func α β)
(μ : measure_theory.measure α) :
Prop
A simple_func has finite measure support if it is equal to 0 outside of a set of finite
measure.
theorem
measure_theory.simple_func.fin_meas_supp_iff_support
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{f : measure_theory.simple_func α β}
{μ : measure_theory.measure α} :
f.fin_meas_supp μ ↔ ⇑μ (function.support ⇑f) < ⊤
theorem
measure_theory.simple_func.fin_meas_supp_iff
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{f : measure_theory.simple_func α β}
{μ : measure_theory.measure α} :
theorem
measure_theory.simple_func.fin_meas_supp.meas_preimage_singleton_ne_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
(h : f.fin_meas_supp μ)
{y : β}
(hy : y ≠ 0) :
theorem
measure_theory.simple_func.fin_meas_supp.map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_zero β]
[has_zero γ]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
{g : β → γ}
(hf : f.fin_meas_supp μ)
(hg : g 0 = 0) :
theorem
measure_theory.simple_func.fin_meas_supp.of_map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_zero β]
[has_zero γ]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
{g : β → γ}
(h : (measure_theory.simple_func.map g f).fin_meas_supp μ)
(hg : ∀ (b : β), g b = 0 → b = 0) :
f.fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.map_iff
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_zero β]
[has_zero γ]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
{g : β → γ}
(hg : ∀ {b : β}, g b = 0 ↔ b = 0) :
(measure_theory.simple_func.map g f).fin_meas_supp μ ↔ f.fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.pair
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_zero β]
[has_zero γ]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
{g : measure_theory.simple_func α γ}
(hf : f.fin_meas_supp μ)
(hg : g.fin_meas_supp μ) :
(f.pair g).fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.map₂
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{δ : Type u_4}
[measurable_space α]
[has_zero β]
[has_zero γ]
[has_zero δ]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
(hf : f.fin_meas_supp μ)
{g : measure_theory.simple_func α γ}
(hg : g.fin_meas_supp μ)
{op : β → γ → δ}
(H : op 0 0 = 0) :
(measure_theory.simple_func.map (function.uncurry op) (f.pair g)).fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.add
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{β : Type u_2}
[add_monoid β]
{f g : measure_theory.simple_func α β}
(hf : f.fin_meas_supp μ)
(hg : g.fin_meas_supp μ) :
(f + g).fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.mul
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{β : Type u_2}
[monoid_with_zero β]
{f g : measure_theory.simple_func α β}
(hf : f.fin_meas_supp μ)
(hg : g.fin_meas_supp μ) :
(f * g).fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.lintegral_lt_top
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α ennreal}
(hm : f.fin_meas_supp μ)
(hf : ∀ᵐ (a : α) ∂μ, ⇑f a < ⊤) :
theorem
measure_theory.simple_func.fin_meas_supp.of_lintegral_lt_top
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α ennreal}
(h : f.lintegral μ < ⊤) :
f.fin_meas_supp μ
theorem
measure_theory.simple_func.fin_meas_supp.iff_lintegral_lt_top
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α ennreal}
(hf : ∀ᵐ (a : α) ∂μ, ⇑f a < ⊤) :
f.fin_meas_supp μ ↔ f.lintegral μ < ⊤
theorem
measure_theory.simple_func.induction
{α : Type u_1}
{γ : Type u_2}
[measurable_space α]
[add_monoid γ]
{P : measure_theory.simple_func α γ → Prop}
(h_ind : ∀ (c : γ) {s : set α} (hs : is_measurable s), P (measure_theory.simple_func.piecewise s hs (measure_theory.simple_func.const α c) (measure_theory.simple_func.const α 0)))
(h_sum : ∀ ⦃f g : measure_theory.simple_func α γ⦄, set.univ ⊆ ⇑f ⁻¹' {0} ∪ ⇑g ⁻¹' {0} → P f → P g → P (f + g))
(f : measure_theory.simple_func α γ) :
P f
To prove something for an arbitrary simple function, it suffices to show that the property holds for (multiples of) characteristic functions and is closed under addition (of functions with disjoint support).
It is possible to make the hypotheses in h_sum a bit stronger, and such conditions can be added
once we need them (for example it is only necessary to consider the case where g is a multiple
of a characteristic function, and that this multiple doesn't appear in the image of f)
def
measure_theory.lintegral
{α : Type u_1}
[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
theorem
measure_theory.simple_func.lintegral_eq_lintegral
{α : Type u_1}
[measurable_space α]
(f : measure_theory.simple_func α ennreal)
(μ : measure_theory.measure α) :
theorem
measure_theory.lintegral_mono'
{α : Type u_1}
[measurable_space α]
⦃μ ν : measure_theory.measure α⦄
(hμν : μ ≤ ν)
⦃f g : α → ennreal⦄
(hfg : f ≤ g) :
theorem
measure_theory.lintegral_mono
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
⦃f g : α → ennreal⦄
(hfg : f ≤ g) :
theorem
measure_theory.monotone_lintegral
{α : Type u_1}
[measurable_space α]
(μ : measure_theory.measure α) :
@[simp]
theorem
measure_theory.lintegral_const
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(c : ennreal) :
theorem
measure_theory.set_lintegral_one
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(s : set α) :
theorem
measure_theory.lintegral_eq_nnreal
{α : Type u_1}
[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 ennreal such that φ ≤ f. This lemma says that it suffices to take
functions φ : α →ₛ ℝ≥0.
theorem
measure_theory.supr_lintegral_le
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{ι : Sort u_2}
(f : ι → α → ennreal) :
theorem
measure_theory.supr2_lintegral_le
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{ι : Sort u_2}
{ι' : ι → Sort u_3}
(f : Π (i : ι), ι' i → α → ennreal) :
theorem
measure_theory.le_infi_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{ι : Sort u_2}
(f : ι → α → ennreal) :
theorem
measure_theory.le_infi2_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{ι : Sort u_2}
{ι' : ι → Sort u_3}
(f : Π (i : ι), ι' i → α → ennreal) :
theorem
measure_theory.lintegral_supr
{α : Type u_1}
[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_eq_supr_eapprox_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_add
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hf : measurable f)
(hg : measurable g) :
theorem
measure_theory.lintegral_zero
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α} :
theorem
measure_theory.lintegral_smul_measure
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(c : ennreal)
(f : α → ennreal) :
theorem
measure_theory.lintegral_sum_measure
{α : Type u_1}
[measurable_space α]
{ι : Type u_2}
(f : α → ennreal)
(μ : ι → measure_theory.measure α) :
@[simp]
theorem
measure_theory.lintegral_zero_measure
{α : Type u_1}
[measurable_space α]
(f : α → ennreal) :
theorem
measure_theory.lintegral_finset_sum
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
(s : finset β)
{f : β → α → ennreal}
(hf : ∀ (b : β), measurable (f b)) :
theorem
measure_theory.lintegral_const_mul
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(r : ennreal)
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_const_mul_le
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(r : ennreal)
(f : α → ennreal) :
theorem
measure_theory.lintegral_mono_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ennreal}
(h : ∀ᵐ (a : α) ∂μ, f a ≤ g a) :
theorem
measure_theory.lintegral_congr_ae
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ennreal}
(h : f =ᵐ[μ] g) :
theorem
measure_theory.lintegral_congr
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f g : α → ennreal}
(h : ∀ (a : α), f a = g a) :
theorem
measure_theory.lintegral_rw₁
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
{f f' : α → β}
(h : f =ᵐ[μ] f')
(g : β → ennreal) :
theorem
measure_theory.lintegral_rw₂
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[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}
[measurable_space α]
{μ : measure_theory.measure α}
(f : α → ennreal)
{s : set α}
(hs : is_measurable s) :
theorem
measure_theory.mul_meas_ge_le_lintegral
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
(ε : ennreal) :
Chebyshev's inequality
theorem
measure_theory.meas_ge_le_lintegral_div
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f)
{ε : ennreal}
(hε : ε ≠ 0)
(hε' : ε ≠ ⊤) :
theorem
measure_theory.lintegral_eq_zero_iff
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : measurable f) :
theorem
measure_theory.lintegral_supr_ae
{α : Type u_1}
[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_infi_ae
{α : Type u_1}
[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}
[measurable_space α]
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h_meas : ∀ (n : ℕ), measurable (f n))
(h_mono : ∀ ⦃m n : ℕ⦄, m ≤ n → f n ≤ f m)
(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}
[measurable_space α]
{μ : measure_theory.measure α}
{f : ℕ → α → ennreal}
(h_meas : ∀ (n : ℕ), measurable (f n)) :
Known as Fatou's lemma
theorem
measure_theory.limsup_lintegral_le
{α : Type u_1}
[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 < ⊤) :
theorem
measure_theory.tendsto_lintegral_of_dominated_convergence
{α : Type u_1}
[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 (𝓝 (f a))) :
filter.tendsto (λ (n : ℕ), ∫⁻ (a : α), F n a ∂μ) filter.at_top (𝓝 (∫⁻ (a : α), f a ∂μ))
Dominated convergence theorem for nonnegative functions
theorem
measure_theory.tendsto_lintegral_filter_of_dominated_convergence
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
{ι : Type u_2}
{l : filter ι}
{F : ι → α → ennreal}
{f : α → ennreal}
(bound : α → ennreal)
(hl_cb : l.is_countably_generated)
(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 (𝓝 (f a))) :
Dominated convergence theorem for filters with a countable basis
theorem
measure_theory.lintegral_supr_directed
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[encodable β]
{f : β → α → ennreal}
(hf : ∀ (b : β), measurable (f b))
(h_directed : directed has_le.le f) :
Monotone convergence for a suprema over a directed family and indexed by an encodable type
theorem
measure_theory.lintegral_tsum
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[encodable β]
{f : β → α → ennreal}
(hf : ∀ (i : β), measurable (f i)) :
theorem
measure_theory.lintegral_Union
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[encodable β]
{s : β → set α}
(hm : ∀ (i : β), is_measurable (s i))
(hd : pairwise (disjoint on s))
(f : α → ennreal) :
theorem
measure_theory.lintegral_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[measurable_space β]
{f : β → ennreal}
{g : α → β}
(hf : measurable f)
(hg : measurable g) :
theorem
measure_theory.set_lintegral_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{μ : measure_theory.measure α}
[measurable_space β]
{f : β → ennreal}
{g : α → β}
{s : set β}
(hs : is_measurable s)
(hf : measurable f)
(hg : measurable g) :
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
def
measure_theory.measure.with_density
{α : Type u_1}
[measurable_space α]
(μ : measure_theory.measure α)
(f : α → ennreal) :
Given a measure μ : measure α and a function f : α → ennreal, 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 : is_measurable s), ∫⁻ (a : α) in s, f a ∂μ) _ _
theorem
measure_theory.with_density_apply
{α : Type u_1}
[measurable_space α]
{μ : measure_theory.measure α}
(f : α → ennreal)
{s : set α}
(hs : is_measurable s) :
theorem
measurable.ennreal_induction
{α : Type u_1}
[measurable_space α]
{P : (α → ennreal) → Prop}
(h_ind : ∀ (c : ennreal) ⦃s : set α⦄, is_measurable s → P (s.indicator (λ (_x : α), c)))
(h_sum : ∀ ⦃f g : α → ennreal⦄, set.univ ⊆ f ⁻¹' {0} ∪ g ⁻¹' {0} → measurable f → measurable g → P f → P g → P (f + g))
(h_supr : ∀ ⦃f : ℕ → α → ennreal⦄, (∀ (n : ℕ), measurable (f n)) → monotone f → (∀ (n : ℕ), P (f n)) → P (λ (x : α), ⨆ (n : ℕ), f n x))
⦃f : α → ennreal⦄
(hf : measurable f) :
P f
To prove something for an arbitrary measurable function into ennreal, it suffices to show
that the property holds for (multiples of) characteristic functions and is closed under addition
and supremum of increasing sequences of functions.
It is possible to make the hypotheses in the induction steps a bit stronger, and such conditions
can be added once we need them (for example in h_sum it is only necessary to consider the sum of
a simple function with a multiple of a characteristic function and that the intersection
of their images is a subset of {0}.