Simple functions #
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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. In this file, we define simple functions and establish their
basic properties; and we construct a sequence of simple functions approximating an arbitrary Borel
measurable function f : α → ℝ≥0∞.
The theorem measurable.ennreal_induction shows that in order to prove something for an arbitrary
measurable function into ℝ≥0∞, it is sufficient to show that the property holds for (multiples of)
characteristic functions and is closed under addition and supremum of increasing sequences of
functions.
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_smul
- 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.simple_func_bot'
{β : Type u_2}
{α : Type u_1}
[nonempty β]
(f : measure_theory.simple_func α β) :
∃ (c : β), f = measure_theory.simple_func.const α c
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)}
@[measurability]
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, measurability]
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, measurability]
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_smul
{α : Type u_1}
{β : Type u_2}
[measurable_space α]
{K : Type u_5}
[has_smul K β] :
has_smul K (measure_theory.simple_func α β)
Equations
- measure_theory.simple_func.has_smul = {smul := λ (k : K) (f : measure_theory.simple_func α β), measure_theory.simple_func.map (has_smul.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_smul 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_smul 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_smul K β]
(k : K)
(f : measure_theory.simple_func α β) :
k • f = measure_theory.simple_func.map (has_smul.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_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
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) :
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) :
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) :
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) :
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)
theorem
measurable.ennreal_induction
{α : Type u_1}
[measurable_space α]
{P : (α → ennreal) → Prop}
(h_ind : ∀ (c : ennreal) ⦃s : set α⦄, measurable_set s → P (s.indicator (λ (_x : α), c)))
(h_add : ∀ ⦃f g : α → ennreal⦄, disjoint (function.support f) (function.support g) → 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 ℝ≥0∞, 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_add 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}.