mathlib documentation

data.​indicator_function

data.​indicator_function

source
add_monoid_hom.​map_indicator
set.​eq_on_indicator
set.​indicator
set.​indicator_add
set.​indicator_apply
set.​indicator_comp_of_zero
set.​indicator_compl
set.​indicator_compl_add_self
set.​indicator_compl_add_self_apply
set.​indicator_congr
set.​indicator_empty
set.​indicator_finset_bUnion
set.​indicator_finset_sum
set.​indicator_indicator
set.​indicator_le
set.​indicator_le'
set.​indicator_le_indicator
set.​indicator_le_indicator_of_subset
set.​indicator_le_self
set.​indicator_le_self'
set.​indicator_mul
set.​indicator_neg
set.​indicator_nonneg
set.​indicator_nonneg'
set.​indicator_nonpos
set.​indicator_nonpos'
set.​indicator_of_mem
set.​indicator_of_not_mem
set.​indicator_preimage
set.​indicator_preimage_of_not_mem
set.​indicator_range_comp
set.​indicator_rel_indicator
set.​indicator_self_add_compl
set.​indicator_self_add_compl_apply
set.​indicator_smul
set.​indicator_sub
set.​indicator_union_of_disjoint
set.​indicator_union_of_not_mem_inter
set.​indicator_univ
set.​indicator_zero
set.​is_add_group_hom.​indicator
set.​is_add_monoid_hom.​indicator
set.​mem_of_indicator_ne_zero
set.​mem_range_indicator
set.​sum_indicator_subset
set.​sum_indicator_subset_of_eq_zero
set.​support_indicator

Indicator function

indicator (s : set α) (f : α → β) (a : α) is f a if a ∈ s and is 0 otherwise.

Implementation note

In mathematics, an indicator function or a characteristic function is a function used to indicate membership of an element in a set s, having the value 1 for all elements of s and the value 0 otherwise. But since it is usually used to restrict a function to a certain set s, we let the indicator function take the value f x for some function f, instead of 1. If the usual indicator function is needed, just set f to be the constant function λx, 1.

Tags

indicator, characteristic

source
def set.​indicator {α : Type u} {β : Type v} [has_zero β] :
set α → (α → β) → α → β

indicator s f a is f a if a ∈ s, 0 otherwise.

Equations
source
theorem set.​indicator_apply {α : Type u} {β : Type v} [has_zero β] (s : set α) (f : α → β) (a : α) :
s.indicator f a = ite (a ∈ s) (f a) 0

source
@[simp]
theorem set.​indicator_of_mem {α : Type u} {β : Type v} [has_zero β] {s : set α} {a : α} (h : a ∈ s) (f : α → β) :
s.indicator f a = f a

source
@[simp]
theorem set.​indicator_of_not_mem {α : Type u} {β : Type v} [has_zero β] {s : set α} {a : α} (h : a ∉ s) (f : α → β) :
s.indicator f a = 0

source
theorem set.​mem_of_indicator_ne_zero {α : Type u} {β : Type v} [has_zero β] {s : set α} {f : α → β} {a : α} :
s.indicator f a ≠ 0 → a ∈ s

If an indicator function is nonzero at a point, that point is in the set.

source
theorem set.​eq_on_indicator {α : Type u} {β : Type v} [has_zero β] {s : set α} {f : α → β} :
set.eq_on (s.indicator f) f s

source
theorem set.​support_indicator {α : Type u} {β : Type v} [has_zero β] {s : set α} {f : α → β} :
function.support (s.indicator f) ⊆ s

source
@[simp]
theorem set.​indicator_range_comp {α : Type u} {β : Type v} [has_zero β] {ι : Sort u_1} (f : ι → α) (g : α → β) :
(set.range f).indicator g ∘ f = g ∘ f

source
theorem set.​indicator_congr {α : Type u} {β : Type v} [has_zero β] {s : set α} {f g : α → β} :
(∀ (a : α), a ∈ s → f a = g a) → s.indicator f = s.indicator g

source
@[simp]
theorem set.​indicator_univ {α : Type u} {β : Type v} [has_zero β] (f : α → β) :
set.univ.indicator f = f

source
@[simp]
theorem set.​indicator_empty {α : Type u} {β : Type v} [has_zero β] (f : α → β) :
∅.indicator f = λ (a : α), 0

source
@[simp]
theorem set.​indicator_zero {α : Type u} (β : Type v) [has_zero β] (s : set α) :
s.indicator (λ (x : α), 0) = λ (x : α), 0

source
theorem set.​indicator_indicator {α : Type u} {β : Type v} [has_zero β] (s t : set α) (f : α → β) :
s.indicator (t.indicator f) = (s ∩ t).indicator f

source
theorem set.​indicator_comp_of_zero {α : Type u} {β : Type v} [has_zero β] {s : set α} {f : α → β} {γ : Type u_1} [has_zero γ] {g : β → γ} :
g 0 = 0 → s.indicator (g ∘ f) = g ∘ s.indicator f

source
theorem set.​indicator_preimage {α : Type u} {β : Type v} [has_zero β] (s : set α) (f : α → β) (B : set β) :
s.indicator f ⁻¹' B = s ∩ f ⁻¹' B ∪ sᶜ ∩ (λ (a : α), 0) ⁻¹' B

source
theorem set.​indicator_preimage_of_not_mem {α : Type u} {β : Type v} [has_zero β] (s : set α) (f : α → β) {t : set β} :
0 ∉ t → s.indicator f ⁻¹' t = s ∩ f ⁻¹' t

source
theorem set.​mem_range_indicator {α : Type u} {β : Type v} [has_zero β] {r : β} {s : set α} {f : α → β} :
r ∈ set.range (s.indicator f) ↔ r = 0 ∧ s ≠ set.univ ∨ r ∈ f '' s

source
theorem set.​indicator_rel_indicator {α : Type u} {β : Type v} [has_zero β] {s : set α} {f g : α → β} {a : α} {r : β → β → Prop} :
r 0 0 → (a ∈ s → r (f a) (g a)) → r (s.indicator f a) (s.indicator g a)

source
theorem set.​sum_indicator_subset_of_eq_zero {α : Type u} {β : Type v} [has_zero β] {γ : Type u_1} [add_comm_monoid γ] (f : α → β) (g : α → β → γ) {s₁ s₂ : finset α} :
s₁ ⊆ s₂ → (∀ (a : α), g a 0 = 0) → s₁.sum (λ (i : α), g i (f i)) = s₂.sum (λ (i : α), g i (↑s₁.indicator f i))

Consider a sum of g i (f i) over a finset. Suppose g is a function such as multiplication, which maps a second argument of 0 to

  1. (A typical use case would be a weighted sum of f i * h i or f i • h i, where f gives the weights that are multiplied by some other function h.) Then if f is replaced by the corresponding indicator function, the finset may be replaced by a possibly larger finset without changing the value of the sum.
source
theorem set.​sum_indicator_subset {α : Type u} {γ : Type u_1} [add_comm_monoid γ] (f : α → γ) {s₁ s₂ : finset α} :
s₁ ⊆ s₂ → s₁.sum (λ (i : α), f i) = s₂.sum (λ (i : α), ↑s₁.indicator f i)

Summing an indicator function over a possibly larger finset is the same as summing the original function over the original finset.

source
theorem set.​indicator_union_of_not_mem_inter {α : Type u} {β : Type v} [add_monoid β] {s t : set α} {a : α} (h : a ∉ s ∩ t) (f : α → β) :
(s ∪ t).indicator f a = s.indicator f a + t.indicator f a

source
theorem set.​indicator_union_of_disjoint {α : Type u} {β : Type v} [add_monoid β] {s t : set α} (h : disjoint s t) (f : α → β) :
(s ∪ t).indicator f = λ (a : α), s.indicator f a + t.indicator f a

source
theorem set.​indicator_add {α : Type u} {β : Type v} [add_monoid β] (s : set α) (f g : α → β) :
s.indicator (λ (a : α), f a + g a) = λ (a : α), s.indicator f a + s.indicator g a

source
@[simp]
theorem set.​indicator_compl_add_self_apply {α : Type u} {β : Type v} [add_monoid β] (s : set α) (f : α → β) (a : α) :
sᶜ.indicator f a + s.indicator f a = f a

source
@[simp]
theorem set.​indicator_compl_add_self {α : Type u} {β : Type v} [add_monoid β] (s : set α) (f : α → β) :
sᶜ.indicator f + s.indicator f = f

source
@[simp]
theorem set.​indicator_self_add_compl_apply {α : Type u} {β : Type v} [add_monoid β] (s : set α) (f : α → β) (a : α) :
s.indicator f a + sᶜ.indicator f a = f a

source
@[simp]
theorem set.​indicator_self_add_compl {α : Type u} {β : Type v} [add_monoid β] (s : set α) (f : α → β) :
s.indicator f + sᶜ.indicator f = f

source
@[instance]
def set.​is_add_monoid_hom.​indicator {α : Type u} (β : Type v) [add_monoid β] (s : set α) :
is_add_monoid_hom (λ (f : α → β), s.indicator f)

Equations
  • _ = _
source
theorem set.​indicator_smul {α : Type u} {β : Type v} [add_monoid β] {𝕜 : Type u_1} [monoid 𝕜] [distrib_mul_action 𝕜 β] (s : set α) (r : 𝕜) (f : α → β) :
s.indicator (λ (x : α), r • f x) = λ (x : α), r • s.indicator f x

source
@[instance]
def set.​is_add_group_hom.​indicator {α : Type u} (β : Type v) [add_group β] (s : set α) :
is_add_group_hom (λ (f : α → β), s.indicator f)

Equations
source
theorem set.​indicator_neg {α : Type u} {β : Type v} [add_group β] (s : set α) (f : α → β) :
s.indicator (λ (a : α), -f a) = λ (a : α), -s.indicator f a

source
theorem set.​indicator_sub {α : Type u} {β : Type v} [add_group β] (s : set α) (f g : α → β) :
s.indicator (λ (a : α), f a - g a) = λ (a : α), s.indicator f a - s.indicator g a

source
theorem set.​indicator_compl {α : Type u} {β : Type v} [add_group β] (s : set α) (f : α → β) :
sᶜ.indicator f = f - s.indicator f

source
theorem set.​indicator_finset_sum {α : Type u} {β : Type u_1} [add_comm_monoid β] {ι : Type u_2} (I : finset ι) (s : set α) (f : ι → α → β) :
s.indicator (I.sum (λ (i : ι), f i)) = I.sum (λ (i : ι), s.indicator (f i))

source
theorem set.​indicator_finset_bUnion {α : Type u} {β : Type u_1} [add_comm_monoid β] {ι : Type u_2} (I : finset ι) (s : ι → set α) {f : α → β} :
(∀ (i : ι), i ∈ I → ∀ (j : ι), j ∈ I → i ≠ j → s i ∩ s j = ∅) → ((⋃ (i : ι) (H : i ∈ I), s i).indicator f = λ (a : α), I.sum (λ (i : ι), (s i).indicator f a))

source
theorem set.​indicator_mul {α : Type u} {β : Type v} [mul_zero_class β] (s : set α) (f g : α → β) :
s.indicator (λ (a : α), f a * g a) = λ (a : α), s.indicator f a * s.indicator g a

source
theorem set.​indicator_nonneg' {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f : α → β} {a : α} :
(a ∈ s → 0 ≤ f a) → 0 ≤ s.indicator f a

source
theorem set.​indicator_nonneg {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f : α → β} (h : ∀ (a : α), a ∈ s → 0 ≤ f a) (a : α) :
0 ≤ s.indicator f a

source
theorem set.​indicator_nonpos' {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f : α → β} {a : α} :
(a ∈ s → f a ≤ 0) → s.indicator f a ≤ 0

source
theorem set.​indicator_nonpos {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f : α → β} (h : ∀ (a : α), a ∈ s → f a ≤ 0) (a : α) :
s.indicator f a ≤ 0

source
theorem set.​indicator_le' {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f g : α → β} :
(∀ (a : α), a ∈ s → f a ≤ g a) → (∀ (a : α), a ∉ s → 0 ≤ g a) → s.indicator f ≤ g

source
theorem set.​indicator_le_indicator {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f g : α → β} {a : α} :
f a ≤ g a → s.indicator f a ≤ s.indicator g a

source
theorem set.​indicator_le_indicator_of_subset {α : Type u} {β : Type v} [has_zero β] [preorder β] {s t : set α} {f : α → β} (h : s ⊆ t) (hf : ∀ (a : α), 0 ≤ f a) (a : α) :
s.indicator f a ≤ t.indicator f a

source
theorem set.​indicator_le_self' {α : Type u} {β : Type v} [has_zero β] [preorder β] {s : set α} {f : α → β} :
(∀ (x : α), x ∉ s → 0 ≤ f x) → s.indicator f ≤ f

source
theorem set.​indicator_le_self {α : Type u} {β : Type u_1} [canonically_ordered_add_monoid β] (s : set α) (f : α → β) :
s.indicator f ≤ f

source
theorem set.​indicator_le {α : Type u} {β : Type u_1} [canonically_ordered_add_monoid β] {s : set α} {f g : α → β} :
(∀ (a : α), a ∈ s → f a ≤ g a) → s.indicator f ≤ g

source
theorem add_monoid_hom.​map_indicator {α : Type u} {M : Type u_1} {N : Type u_2} [add_monoid M] [add_monoid N] (f : M →+ N) (s : set α) (g : α → M) (x : α) :
⇑f (s.indicator g x) = s.indicator (⇑f ∘ g) x

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string
basic
defs
zmod
basic
zsqrtd
basic
gaussian_int
W
bool
char
dfinsupp
erased
fin
fin2
fin_enum
finmap
hash_map
holor
indicator_function
lazy_list2
mllist
monoid_algebra
mv_polynomial
opposite
pequiv
pfun
prod
quot
rel
semiquot
subtype
sum
support
sym
sym2
tree
typevec
ulift
vector2
deprecated
group
ring
subgroup
submonoid
dynamics
circle
rotation_number
translation_number
fixed_points
basic
topology
periodic_pts
field_theory
algebraic_closure
chevalley_warning
finite
fixed
minimal_polynomial
mv_polynomial
normal
perfect_closure
separable
splitting_field
subfield
tower
geometry
algebra
lie_group
euclidean
basic
circumcenter
monge_point
triangle
manifold
basic_smooth_bundle
charted_space
local_invariant_properties
mfderiv
real_instances
smooth_manifold_with_corners
times_cont_mdiff
group_theory
perm
cycles
sign
submonoid
basic
membership
operations
abelianization
congruence
coset
eckmann_hilton
free_abelian_group
free_group
group_action
monoid_localization
order_of_element
presented_group
quotient_group
semidirect_product
subgroup
sylow
linear_algebra
affine_space
basic
combination
finite_dimensional
independent
char_poly
coeff
direct_sum
finsupp
tensor_product
adic_completion
basic
basis
bilinear_form
char_poly
contraction
determinant
dimension
direct_sum_module
dual
finite_dimensional
finsupp
finsupp_vector_space
invariant_basis_number
lagrange
linear_action
linear_pmap
matrix
multilinear
nonsingular_inverse
projection
quadratic_form
sesquilinear_form
smodeq
special_linear_group
tensor_algebra
tensor_product
logic
function
basic
conjugate
iterate
basic
embedding
nontrivial
relation
relator
unique
measure_theory
category
Meas
ae_eq_fun
bochner_integration
borel_space
content
decomposition
giry_monad
group
haar_measure
integration
interval_integral
l1_space
lebesgue_measure
measurable_space
measure_space
outer_measure
probability_mass_function
set_integral
simple_func_dense
meta
coinductive_predicates
expr
expr_lens
rb_map
uchange
number_theory
basic
bernoulli
dioph
lucas_lehmer
pell
primorial
pythagorean_triples
quadratic_reciprocity
sum_four_squares
sum_two_squares
order
category
LinearOrder
NonemptyFinLinOrd
PartialOrder
Preorder
filter
at_top_bot
bases
basic
cofinite
countable_Inter
extr
filter_product
germ
indicator_function
interval
lift
partial
pointwise
ultrafilter
basic
boolean_algebra
bounded_lattice
bounds
complete_boolean_algebra
complete_lattice
conditionally_complete_lattice
copy
directed
fixed_points
galois_connection
ideal
iterate
lattice
lexicographic
liminf_limsup
ord_continuous
order_iso_nat
pilex
rel_classes
rel_iso
semiconj_Sup
zorn
representation_theory
maschke
ring_theory
ideal
basic
operations
over
polynomial
basic
homogeneous
rational_root
valuation
basic
adjoin
adjoin_root
algebra
algebra_operations
algebra_tower
algebraic
coprime
derivation
discrete_valuation_ring
eisenstein_criterion
euclidean_domain
fintype
fractional_ideal
free_comm_ring
free_ring
integral_closure
integral_domain
jacobson
jacobson_ideal
localization
maps
matrix_algebra
multiplicity
noetherian
polynomial_algebra
power_series
prime
principal_ideal_domain
subring
subsemiring
tensor_product
unique_factorization_domain
set_theory
game
domineering
impartial
nim
short
state
winner
cardinal
cardinal_ordinal
cofinality
game
lists
ordinal
ordinal_arithmetic
ordinal_notation
pgame
schroeder_bernstein
surreal
zfc
system
random
basic
tactic
converter
apply_congr
binders
interactive
old_conv
linarith
datatypes
elimination
frontend
lemmas
parsing
preprocessing
verification
lint
basic
default
frontend
misc
simp
type_classes
monotonicity
basic
interactive
lemmas
nth_rewrite
basic
congr
default
omega
int
dnf
form
main
preterm
nat
dnf
form
main
neg_elim
preterm
sub_elim
clause
coeffs
eq_elim
find_ees
find_scalars
lin_comb
main
misc
prove_unsats
term
rewrite_all
basic
abel
algebra
alias
apply
apply_fun
auto_cases
binder_matching
cache
cancel_denoms
chain
choose
clear
core
dec_trivial
delta_instance
derive_fintype
derive_inhabited
doc_commands
elide
equiv_rw
explode
ext
fin_cases
find
find_unused
finish
fix_reflect_string
generalize_proofs
generalizes
group
hint
interactive
interactive_expr
interval_cases
lift
local_cache
localized
mk_iff_of_inductive_prop
noncomm_ring
norm_cast
norm_num
obviously
pi_instances
protected
push_neg
rcases
reassoc_axiom
rename_var
replacer
restate_axiom
rewrite
ring
ring2
ring_exp
scc
show_term
simp_command
simp_result
simp_rw
simpa
simps
slice
solve_by_elim
split_ifs
squeeze
subtype_instance
suggest
tauto
tfae
tidy
transfer
transform_decl
transport
trunc_cases
unfold_cases
where
with_local_reducibility
wlog
zify
topology
algebra
affine
continuous_functions
group
group_completion
infinite_sum
module
monoid
multilinear
open_subgroup
ordered
polynomial
ring
uniform_group
uniform_ring
category
Top
adjunctions
basic
epi_mono
limits
open_nhds
opens
TopCommRing
UniformSpace
instances
complex
ennreal
nnreal
real
real_vector_space
metric_space
antilipschitz
baire
basic
cau_seq_filter
closeds
completion
contracting
emetric_space
gluing
gromov_hausdorff
gromov_hausdorff_realized
hausdorff_distance
isometry
lipschitz
pi_Lp
premetric_space
sheaves
presheaf
presheaf_of_functions
sheaf
sheaf_of_functions
stalks
uniform_space
absolute_value
abstract_completion
basic
cauchy
compact_separated
compare_reals
complete_separated
completion
pi
separation
uniform_convergence
uniform_embedding
bases
basic
bounded_continuous_function
compact_open
compacts
constructions
continuous_map
continuous_on
dense_embedding
extend_from_subset
homeomorph
list
local_extr
local_homeomorph
maps
opens
order
path_connected
separation
sequences
stone_cech
subset_properties
tactic
topological_fiber_bundle