Lebesgue integral for ℝ≥0∞-valued functions #
We define simple functions and show that each Borel measurable function on ℝ≥0∞ can be
approximated by a sequence of simple functions.
To prove something for an arbitrary measurable function into ℝ≥0∞, 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 → 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).
structure
measure_theory.simple_func
(α : Type u)
[measurable_space α]
(β : Type v) :
Type (max u v)
- to_fun : α → β
- measurable_set_fiber' : ∀ (x : β), measurable_set (self.to_fun ⁻¹' {x})
- finite_range' : (set.range self.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.
Instances for measure_theory.simple_func
- measure_theory.simple_func.has_sizeof_inst
- measure_theory.simple_func.has_coe_to_fun
- measure_theory.simple_func.inhabited
- measure_theory.simple_func.has_one
- measure_theory.simple_func.has_zero
- measure_theory.simple_func.has_mul
- measure_theory.simple_func.has_add
- measure_theory.simple_func.has_div
- measure_theory.simple_func.has_sub
- measure_theory.simple_func.has_inv
- measure_theory.simple_func.has_neg
- measure_theory.simple_func.has_sup
- measure_theory.simple_func.has_inf
- measure_theory.simple_func.has_le
- measure_theory.simple_func.has_scalar
- measure_theory.simple_func.has_nat_pow
- measure_theory.simple_func.has_int_pow
- measure_theory.simple_func.add_monoid
- measure_theory.simple_func.add_comm_monoid
- measure_theory.simple_func.add_group
- measure_theory.simple_func.add_comm_group
- measure_theory.simple_func.monoid
- measure_theory.simple_func.comm_monoid
- measure_theory.simple_func.group
- measure_theory.simple_func.comm_group
- measure_theory.simple_func.module
- measure_theory.simple_func.preorder
- measure_theory.simple_func.partial_order
- measure_theory.simple_func.order_bot
- measure_theory.simple_func.order_top
- measure_theory.simple_func.semilattice_inf
- measure_theory.simple_func.semilattice_sup
- measure_theory.simple_func.lattice
- measure_theory.simple_func.bounded_order
@[protected, instance]
def
measure_theory.simple_func.has_coe_to_fun
{α : Type u_1}
{β : Type u_2}
[measurable_space α] :
has_coe_to_fun (measure_theory.simple_func α β) (λ (_x : measure_theory.simple_func α β), α → 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.measurable_set_fiber
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(x : β) :
measurable_set (⇑f ⁻¹' {x})
@[simp]
theorem
measure_theory.simple_func.apply_mk
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : α → β)
(h : ∀ (x : β), measurable_set (f ⁻¹' {x}))
(h' : (set.range f).finite)
(x : α) :
⇑{to_fun := f, measurable_set_fiber' := h, finite_range' := h'} x = f x
def
measure_theory.simple_func.of_is_empty
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[is_empty α] :
Simple function defined on the empty type.
Equations
- measure_theory.simple_func.of_is_empty = {to_fun := is_empty_elim (λ (a : α), β), measurable_set_fiber' := _, finite_range' := _}
@[protected]
noncomputable
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) :
theorem
measure_theory.simple_func.exists_forall_le
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[nonempty β]
[preorder β]
[is_directed β has_le.le]
(f : measure_theory.simple_func α β) :
Constant function as a simple_func.
Equations
- measure_theory.simple_func.const α b = {to_fun := λ (a : α), b, measurable_set_fiber' := _, finite_range' := _}
@[protected, instance]
def
measure_theory.simple_func.inhabited
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[inhabited β] :
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.range_const_subset
{β : Type u_2}
(α : Type u_1)
[measurable_space α]
(b : β) :
(measure_theory.simple_func.const α b).range ⊆ {b}
theorem
measure_theory.simple_func.measurable_set_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), measurable_set {a : α | r a b}) :
measurable_set {a : α | r a (⇑f a)}
theorem
measure_theory.simple_func.measurable_set_preimage
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(f : measure_theory.simple_func α β)
(s : set β) :
measurable_set (⇑f ⁻¹' s)
@[protected]
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
@[protected]
theorem
measure_theory.simple_func.ae_measurable
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[measurable_space β]
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α β) :
ae_measurable ⇑f μ
@[protected]
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) :
noncomputable
def
measure_theory.simple_func.piecewise
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
(s : set α)
(hs : measurable_set 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), classical.prop_decidable (j ∈ s)), measurable_set_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 : measurable_set 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 : measurable_set 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 : measurable_set 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.support_indicator
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β]
{s : set α}
(hs : measurable_set s)
(f : measure_theory.simple_func α β) :
theorem
measure_theory.simple_func.range_indicator
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{s : set α}
(hs : measurable_set s)
(hs_nonempty : s.nonempty)
(hs_ne_univ : s ≠ set.univ)
(x y : β) :
(measure_theory.simple_func.piecewise s hs (measure_theory.simple_func.const α x) (measure_theory.simple_func.const α y)).range = {x, y}
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, measurable_set_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, measurable_set_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) :
noncomputable
def
measure_theory.simple_func.extend
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f₁ : measure_theory.simple_func α γ)
(g : α → β)
(hg : measurable_embedding g)
(f₂ : measure_theory.simple_func β γ) :
Extend a simple_func along a measurable embedding: f₁.extend g hg f₂ is the function
F : β →ₛ γ such that F ∘ g = f₁ and F y = f₂ y whenever y ∉ range g.
Equations
- f₁.extend g hg f₂ = {to_fun := function.extend g ⇑f₁ ⇑f₂, measurable_set_fiber' := _, finite_range' := _}
@[simp]
theorem
measure_theory.simple_func.extend_apply
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f₁ : measure_theory.simple_func α γ)
{g : α → β}
(hg : measurable_embedding g)
(f₂ : measure_theory.simple_func β γ)
(x : α) :
@[simp]
theorem
measure_theory.simple_func.extend_apply'
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f₁ : measure_theory.simple_func α γ)
{g : α → β}
(hg : measurable_embedding g)
(f₂ : measure_theory.simple_func β γ)
{y : β}
(h : ¬∃ (x : α), g x = y) :
@[simp]
theorem
measure_theory.simple_func.extend_comp_eq'
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f₁ : measure_theory.simple_func α γ)
{g : α → β}
(hg : measurable_embedding g)
(f₂ : measure_theory.simple_func β γ) :
@[simp]
theorem
measure_theory.simple_func.extend_comp_eq
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[measurable_space β]
(f₁ : measure_theory.simple_func α γ)
{g : α → β}
(hg : measurable_embedding g)
(f₂ : measure_theory.simple_func β γ) :
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
@[protected, instance]
def
measure_theory.simple_func.has_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β] :
Equations
@[protected, instance]
def
measure_theory.simple_func.has_one
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_one β] :
Equations
@[protected, 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}
@[protected, 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}
@[protected, instance]
def
measure_theory.simple_func.has_div
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_div β] :
Equations
- measure_theory.simple_func.has_div = {div := λ (f g : measure_theory.simple_func α β), (measure_theory.simple_func.map has_div.div f).seq g}
@[protected, instance]
def
measure_theory.simple_func.has_sub
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sub β] :
Equations
- measure_theory.simple_func.has_sub = {sub := λ (f g : measure_theory.simple_func α β), (measure_theory.simple_func.map has_sub.sub f).seq g}
@[protected, instance]
def
measure_theory.simple_func.has_inv
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_inv β] :
Equations
@[protected, instance]
def
measure_theory.simple_func.has_neg
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_neg β] :
Equations
@[protected, 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}
@[protected, 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}
@[protected, 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.const_one
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_one β] :
@[simp]
theorem
measure_theory.simple_func.const_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β] :
@[simp, norm_cast]
theorem
measure_theory.simple_func.coe_one
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_one β] :
@[simp, norm_cast]
theorem
measure_theory.simple_func.coe_zero
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_zero β] :
@[simp, norm_cast]
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, norm_cast]
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, norm_cast]
theorem
measure_theory.simple_func.coe_inv
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_inv β]
(f : measure_theory.simple_func α β) :
@[simp, norm_cast]
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) :
@[simp, norm_cast]
theorem
measure_theory.simple_func.coe_div
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_div β]
(f g : measure_theory.simple_func α β) :
@[simp, norm_cast]
theorem
measure_theory.simple_func.coe_sub
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sub β]
(f g : measure_theory.simple_func α β) :
@[simp, norm_cast]
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, norm_cast]
theorem
measure_theory.simple_func.coe_sup
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sup β]
(f g : measure_theory.simple_func α β) :
@[simp, norm_cast]
theorem
measure_theory.simple_func.coe_inf
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_inf β]
(f g : measure_theory.simple_func α β) :
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.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.div_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_div β]
(f g : measure_theory.simple_func α β)
(x : α) :
theorem
measure_theory.simple_func.sub_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_sub β]
(f g : measure_theory.simple_func α β)
(x : α) :
theorem
measure_theory.simple_func.neg_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_neg β]
(f : measure_theory.simple_func α β)
(x : α) :
theorem
measure_theory.simple_func.inv_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_inv β]
(f : measure_theory.simple_func α β)
(x : α) :
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.inf_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_inf β]
(f g : measure_theory.simple_func α β)
(a : α) :
@[simp]
theorem
measure_theory.simple_func.range_one
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[nonempty α]
[has_one 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] :
@[simp]
theorem
measure_theory.simple_func.range_eq_empty_of_is_empty
{α : Type u_1}
[measurable_space α]
{β : Type u_2}
[hα : is_empty α]
(f : measure_theory.simple_func α 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} :
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
theorem
measure_theory.simple_func.const_add_eq_map
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_add 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
theorem
measure_theory.simple_func.map_add
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_add β]
[has_add γ]
{g : β → γ}
(hg : ∀ (x y : β), g (x + y) = g x + g y)
(f₁ f₂ : measure_theory.simple_func α β) :
measure_theory.simple_func.map g (f₁ + f₂) = measure_theory.simple_func.map g f₁ + measure_theory.simple_func.map g f₂
theorem
measure_theory.simple_func.map_mul
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
[measurable_space α]
[has_mul β]
[has_mul γ]
{g : β → γ}
(hg : ∀ (x y : β), g (x * y) = g x * g y)
(f₁ f₂ : measure_theory.simple_func α β) :
measure_theory.simple_func.map g (f₁ * f₂) = measure_theory.simple_func.map g f₁ * measure_theory.simple_func.map g f₂
@[protected, 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) :
@[protected, instance]
def
measure_theory.simple_func.has_nat_pow
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[monoid β] :
Equations
- measure_theory.simple_func.has_nat_pow = {pow := λ (f : measure_theory.simple_func α β) (n : ℕ), measure_theory.simple_func.map (λ (_x : β), _x ^ n) f}
@[simp]
theorem
measure_theory.simple_func.coe_pow
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[monoid β]
(f : measure_theory.simple_func α β)
(n : ℕ) :
theorem
measure_theory.simple_func.pow_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[monoid β]
(n : ℕ)
(f : measure_theory.simple_func α β)
(a : α) :
@[protected, instance]
def
measure_theory.simple_func.has_int_pow
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[div_inv_monoid β] :
Equations
- measure_theory.simple_func.has_int_pow = {pow := λ (f : measure_theory.simple_func α β) (n : ℤ), measure_theory.simple_func.map (λ (_x : β), _x ^ n) f}
@[simp]
theorem
measure_theory.simple_func.coe_zpow
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[div_inv_monoid β]
(f : measure_theory.simple_func α β)
(z : ℤ) :
theorem
measure_theory.simple_func.zpow_apply
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[div_inv_monoid β]
(z : ℤ)
(f : measure_theory.simple_func α β)
(a : α) :
@[protected, 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 measure_theory.simple_func.add_monoid._proof_3
@[protected, 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 measure_theory.simple_func.add_comm_monoid._proof_3
@[protected, 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 measure_theory.simple_func.add_group._proof_4 measure_theory.simple_func.add_group._proof_5 measure_theory.simple_func.add_group._proof_6
@[protected, 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 measure_theory.simple_func.add_comm_group._proof_4 measure_theory.simple_func.add_comm_group._proof_5 measure_theory.simple_func.add_comm_group._proof_6
@[protected, instance]
def
measure_theory.simple_func.monoid
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[monoid β] :
Equations
- measure_theory.simple_func.monoid = function.injective.monoid (λ (f : measure_theory.simple_func α β), show α → β, from ⇑f) measure_theory.simple_func.coe_injective measure_theory.simple_func.monoid._proof_1 measure_theory.simple_func.monoid._proof_2 measure_theory.simple_func.coe_pow
@[protected, instance]
def
measure_theory.simple_func.comm_monoid
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[comm_monoid β] :
Equations
- measure_theory.simple_func.comm_monoid = function.injective.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.comm_monoid._proof_1 measure_theory.simple_func.comm_monoid._proof_2 measure_theory.simple_func.comm_monoid._proof_3
@[protected, instance]
Equations
- measure_theory.simple_func.group = function.injective.group (λ (f : measure_theory.simple_func α β), show α → β, from ⇑f) measure_theory.simple_func.coe_injective measure_theory.simple_func.group._proof_1 measure_theory.simple_func.group._proof_2 measure_theory.simple_func.group._proof_3 measure_theory.simple_func.group._proof_4 measure_theory.simple_func.group._proof_5 measure_theory.simple_func.group._proof_6
@[protected, instance]
def
measure_theory.simple_func.comm_group
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[comm_group β] :
Equations
- measure_theory.simple_func.comm_group = function.injective.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.comm_group._proof_1 measure_theory.simple_func.comm_group._proof_2 measure_theory.simple_func.comm_group._proof_3 measure_theory.simple_func.comm_group._proof_4 measure_theory.simple_func.comm_group._proof_5 measure_theory.simple_func.comm_group._proof_6
@[protected, instance]
def
measure_theory.simple_func.module
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[semiring K]
[add_comm_monoid β]
[module K β] :
module K (measure_theory.simple_func α β)
Equations
- measure_theory.simple_func.module = function.injective.module 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.module._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
@[protected, 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 := _}
@[protected, instance]
def
measure_theory.simple_func.partial_order
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[partial_order β] :
Equations
@[protected, instance]
def
measure_theory.simple_func.order_bot
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_le β]
[order_bot β] :
Equations
@[protected, instance]
def
measure_theory.simple_func.order_top
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_le β]
[order_top β] :
Equations
@[protected, 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 := _}
@[protected, 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 := _}
@[protected, 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 := _}
@[protected, instance]
def
measure_theory.simple_func.bounded_order
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[has_le β]
[bounded_order β] :
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 β]
[order_bot β]
{f : γ → measure_theory.simple_func α β}
(s : finset γ)
(a : α) :
noncomputable
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 (measurable_set s) (λ (hs : measurable_set s), measure_theory.simple_func.piecewise s hs f 0) (λ (hs : ¬measurable_set 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 : ¬measurable_set 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 : measurable_set 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 nnreal)
(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 nnreal)
(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 : measurable_set 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 : measurable_set 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 : measurable_set 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 : measurable_set 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) :
noncomputable
def
measure_theory.simple_func.approx
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
[semilattice_sup β]
[order_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 = ℝ≥0∞ 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 β]
[order_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 β]
[order_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 β]
[order_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 ℝ≥0∞s such that its range is the set of non-negative rational numbers.
Equations
noncomputable
def
measure_theory.simple_func.eapprox
{α : Type u_1}
[measurable_space α] :
(α → ennreal) → ℕ → measure_theory.simple_func α ennreal
Approximate a function α → ℝ≥0∞ by a sequence of simple functions.
theorem
measure_theory.simple_func.eapprox_lt_top
{α : Type u_1}
[measurable_space α]
(f : α → ennreal)
(n : ℕ)
(a : α) :
⇑(measure_theory.simple_func.eapprox f n) a < ⊤
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
noncomputable
def
measure_theory.simple_func.eapprox_diff
{α : Type u_1}
[measurable_space α]
(f : α → ennreal)
(n : ℕ) :
Approximate a function α → ℝ≥0∞ by a series of simple functions taking their values
in ℝ≥0.
Equations
- measure_theory.simple_func.eapprox_diff f (n + 1) = measure_theory.simple_func.map ennreal.to_nnreal (measure_theory.simple_func.eapprox f (n + 1) - measure_theory.simple_func.eapprox f n)
- measure_theory.simple_func.eapprox_diff f 0 = measure_theory.simple_func.map ennreal.to_nnreal (measure_theory.simple_func.eapprox f 0)
theorem
measure_theory.simple_func.sum_eapprox_diff
{α : Type u_1}
[measurable_space α]
(f : α → ennreal)
(n : ℕ)
(a : α) :
(finset.range (n + 1)).sum (λ (k : ℕ), ↑(⇑(measure_theory.simple_func.eapprox_diff f k) a)) = ⇑(measure_theory.simple_func.eapprox f n) a
theorem
measure_theory.simple_func.tsum_eapprox_diff
{α : Type u_1}
[measurable_space α]
(f : α → ennreal)
(hf : measurable f)
(a : α) :
∑' (n : ℕ), ↑(⇑(measure_theory.simple_func.eapprox_diff f n) a) = f a
noncomputable
def
measure_theory.simple_func.lintegral
{α : Type u_1}
{m : measurable_space α}
(f : measure_theory.simple_func α ennreal)
(μ : measure_theory.measure α) :
Integral of a simple function whose codomain is ℝ≥0∞.
theorem
measure_theory.simple_func.lintegral_eq_of_subset
{α : Type u_1}
{m : 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.lintegral_eq_of_subset'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
{s : finset ennreal}
(hs : f.range \ {0} ⊆ s) :
theorem
measure_theory.simple_func.map_lintegral
{α : Type u_1}
{β : Type u_2}
{m : 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 → ℝ≥0∞ and f : α →ₛ β.
theorem
measure_theory.simple_func.add_lintegral
{α : Type u_1}
{m : 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}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
(x : ennreal) :
Integral of a simple function α →ₛ ℝ≥0∞ as a bilinear map.
@[simp]
theorem
measure_theory.simple_func.zero_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α} :
theorem
measure_theory.simple_func.lintegral_add
{α : Type u_1}
{m : 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}
{m : 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}
{m : 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}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.simple_func.lintegral_restrict
{α : Type u_1}
{m : 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}
{m : measurable_space α}
{μ : measure_theory.measure α}
(f : measure_theory.simple_func α ennreal)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.simple_func.const_lintegral
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(c : ennreal) :
theorem
measure_theory.simple_func.const_lintegral_restrict
{α : Type u_1}
{m : 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}
{m : measurable_space α}
{μ : measure_theory.measure α}
(c : ennreal)
{s : set α}
(hs : measurable_set s) :
theorem
measure_theory.simple_func.le_sup_lintegral
{α : Type u_1}
{m : 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}
{m : measurable_space α}
{μ ν : measure_theory.measure α}
{f g : measure_theory.simple_func α ennreal}
(hfg : f ≤ g)
(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}
{m : 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}
{m : 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}
{m : 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), measurable_set s → ⇑μ' s = ⇑μ (m' ⁻¹' s)) :
theorem
measure_theory.simple_func.lintegral_map
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{β : Type u_2}
[measurable_space β]
(g : measure_theory.simple_func β ennreal)
{f : α → β}
(hf : measurable f) :
g.lintegral (measure_theory.measure.map f μ) = (g.comp f hf).lintegral μ
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}
theorem
measure_theory.simple_func.measurable_set_support
{α : Type u_1}
{β : Type u_2}
[has_zero β]
[measurable_space α]
(f : measure_theory.simple_func α β) :
@[protected]
def
measure_theory.simple_func.fin_meas_supp
{α : Type u_1}
{β : Type u_2}
[has_zero β]
{m : measurable_space α}
(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}
{m : measurable_space α}
[has_zero β]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β} :
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}
{m : measurable_space α}
[has_zero β]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β} :
theorem
measure_theory.simple_func.fin_meas_supp.meas_preimage_singleton_ne_zero
{α : Type u_1}
{β : Type u_2}
{m : 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) :
@[protected]
theorem
measure_theory.simple_func.fin_meas_supp.map
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : 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}
{m : 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}
{m : 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
@[protected]
theorem
measure_theory.simple_func.fin_meas_supp.pair
{α : Type u_1}
{β : Type u_2}
{γ : Type u_3}
{m : 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
@[protected]
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}
{m : measurable_space α}
[has_zero β]
[has_zero γ]
{μ : measure_theory.measure α}
{f : measure_theory.simple_func α β}
[has_zero δ]
(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
@[protected]
theorem
measure_theory.simple_func.fin_meas_supp.add
{α : Type u_1}
{m : 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
@[protected]
theorem
measure_theory.simple_func.fin_meas_supp.mul
{α : Type u_1}
{m : 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}
{m : 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_ne_top
{α : Type u_1}
{m : 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}
{m : 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 < ⊤
@[protected]
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 : measurable_set s), P (measure_theory.simple_func.piecewise s hs (measure_theory.simple_func.const α c) (measure_theory.simple_func.const α 0)))
(h_add : ∀ ⦃f g : measure_theory.simple_func α γ⦄, disjoint (function.support ⇑f) (function.support ⇑g) → 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_add 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)
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 ν : measure_theory.measure α⦄
(hμν : μ ≤ ν)
⦃f g : α → ennreal⦄
(hfg : f ≤ g) :
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.lintegral_mono_nnreal
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → nnreal}
(h : 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.supr_lintegral_le
{α : Type u_1}
{m : 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}
{m : 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}
{m : 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}
{m : 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_mono_ae
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(h : ∀ᵐ (a : α) ∂μ, f a ≤ g a) :
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_congr_ae
{α : Type u_1}
{m : 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}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(h : ∀ (a : α), f a = g a) :
theorem
measure_theory.set_lintegral_congr_fun
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
{s : set α}
(hs : measurable_set s)
(hfg : ∀ᵐ (x : α) ∂μ, x ∈ s → f x = g x) :
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.
theorem
measure_theory.lintegral_add_left'
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(g : α → ennreal) :
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 : ae_measurable g μ) :
@[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_smul_measure
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(c : ennreal)
(f : α → ennreal) :
@[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_finset_sum_measure
{α : Type u_1}
{ι : Type u_2}
{m : measurable_space α}
(s : finset ι)
(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_univ
{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) :
theorem
measure_theory.lintegral_finset_sum'
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : finset β)
{f : β → α → ennreal}
(hf : ∀ (b : β), b ∈ s → ae_measurable (f b) μ) :
theorem
measure_theory.lintegral_finset_sum
{α : Type u_1}
{β : Type u_2}
{m : measurable_space α}
{μ : measure_theory.measure α}
(s : finset β)
{f : β → α → ennreal}
(hf : ∀ (b : β), b ∈ s → measurable (f b)) :
@[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_const_mul_le
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
(f : α → ennreal) :
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_mul_const_le
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
(r : ennreal)
(f : α → ennreal) :
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_rw₁
{α : Type u_1}
{β : Type u_2}
{m : 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}
{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.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) :
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_pos
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f : α → ennreal}
(hf : ae_measurable f μ)
(hμf : 0 < ⇑μ {x : α | f x = ⊤}) :
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 : 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.lintegral_strict_mono
{α : Type u_1}
{m : measurable_space α}
{μ : measure_theory.measure α}
{f g : α → ennreal}
(hμ : μ ≠ 0)
(hg : ae_measurable g μ)
(hfi : ∫⁻ (x : α), f x ∂μ ≠ ⊤)
(h : ∀ᵐ (x : α) ∂μ, 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) :
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)) :
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 ≠ ⊤) :
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 →