Measurable Set Ontology

flow "MeasurableSetOntology" v1.0

# MEASURABLE SET ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/measure-theory/measurable-set-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Measure Theory"
ENTITY := "Measurable Set"

MEASURABLE_SET := {
 type: MeasureTheoreticSetEntity,
 domain: MeasureTheory,
 purpose: "Represent a subset belonging to a specified sigma algebra and therefore eligible for measure-theoretic operations."
}

CONCEPTS := {
 MEASURABLE_SET,
 UNDERLYING_SET,
 SIGMA_ALGEBRA,
 MEASURABLE_SPACE,
 MEASURE,
 NULL_SET,
 BOREL_SET,
 LEBESGUE_MEASURABLE_SET,
 CARATHEODORY_MEASURABLE_SET,
 COMPLETION,
 INDICATOR_FUNCTION,
 SET_LIMIT,
 PRODUCT_MEASURABLE_SET
}

# ---------- DEFINITION ----------
let
 (X,Sigma)
 -> MEASURABLE_SPACE

A subset_of X

A
 -> MEASURABLE_SET

iff

A in Sigma

MEASURABILITY
 -> relative_to(SIGMA_ALGEBRA)

# ---------- STRUCTURE DEPENDENCE ----------
same subset A
 -> may_be_measurable_in(Sigma_1)
 -> may_not_be_measurable_in(Sigma_2)

MEASURABLE_SET
 -/> intrinsic_without(MEASURABLE_STRUCTURE)

# ---------- BASIC MEMBERS ----------
X
 in Sigma

empty_set
 in Sigma

therefore

X
 -> MEASURABLE_SET

empty_set
 -> MEASURABLE_SET

# ---------- COMPLEMENT ----------
A in Sigma

-> X\A in Sigma

MEASURABLE_SET
 -> complement_is(MEASURABLE)

# ---------- COUNTABLE UNION ----------
A_n in Sigma
 for_all(n)

-> union_n(A_n)
   in Sigma

COUNTABLE_UNION_OF_MEASURABLE_SETS
 -> MEASURABLE

# ---------- COUNTABLE INTERSECTION ----------
A_n in Sigma

-> intersection_n(A_n)
   in Sigma

COUNTABLE_INTERSECTION_OF_MEASURABLE_SETS
 -> MEASURABLE

# ---------- DIFFERENCE ----------
A,B in Sigma

A\B
 =
A intersection (X\B)

therefore

A\B
 -> MEASURABLE_SET

# ---------- SYMMETRIC DIFFERENCE ----------
A triangle B
 :=
(A\B)
union
(B\A)

if
 A,B measurable

then

A triangle B
 -> MEASURABLE

# ---------- SET LIMITS ----------
limsup A_n
 :=
intersection_n(
 union_{k>=n} A_k
)

liminf A_n
 :=
union_n(
 intersection_{k>=n} A_k
)

if
 A_n measurable

then

limsup A_n
 and
liminf A_n
 -> MEASURABLE

SIGMA_ALGEBRA
 -> preserves(COUNTABLE_SET_LIMITS)

# ---------- MEASURE ----------
given measure

mu:
 Sigma -> [0,infinity]

if
 A measurable

then

mu(A)
 -> DEFINED

MEASURABLE_SET
 -> belongs_to(DOMAIN_OF_MEASURE)

NONMEASURABLE_SET
 -> ordinary_measure_value_not_defined_without_extension_or_outer_measure_context

# ---------- NULL SET ----------
A
 -> NULL_SET

if

A measurable
and
mu(A)=0

NULL_SET
 -> MEASURABLE_SET

in complete measure space:

B subset_of A
and
mu(A)=0

-> B measurable
-> mu(B)=0

# ---------- ALMOST EVERYWHERE ----------
property P(x)

failure_set:

N
 :=
{x in X : P(x) false}

if
 N measurable
and
mu(N)=0

then

P
 -> holds(ALMOST_EVERYWHERE)

MEASURABLE_NULL_SET
 -> defines(NEGLIGIBLE_EXCEPTION)

# ---------- BOREL SET ----------
for topological space
 (X,Tau)

BOREL_SIGMA_ALGEBRA
 :=
sigma(Tau)

A
 -> BOREL_SET

iff

A in B(X)

BOREL_SET
 -> measurable_relative_to(BOREL_STRUCTURE)

# ---------- LEBESGUE MEASURABLE ----------
on R^n:

LEBESGUE_SIGMA_ALGEBRA
 -> completion_of(BOREL_STRUCTURE_UNDER_LEBESGUE_MEASURE)

A
 -> LEBESGUE_MEASURABLE_SET

iff

A in LEBESGUE_SIGMA_ALGEBRA

BOREL_SET
 -> LEBESGUE_MEASURABLE

converse
 -/> universally_true

# ---------- OUTER MEASURE ----------
outer measure
 mu_star

defined_on
 POWER_SET(X)

A
 -> CARATHEODORY_MEASURABLE

iff

for_all(E subset_of X):

mu_star(E)
 =
mu_star(E intersection A)
+
mu_star(E\A)

CARATHEODORY_MEASURABILITY
 -> constructs(MEASURABLE_SET_CLASS_FROM_OUTER_MEASURE)

# ---------- CARATHEODORY SIGMA ALGEBRA ----------
collection

M
 :=
{
 A subset_of X :
 A satisfies_Caratheodory_condition
}

M
 -> SIGMA_ALGEBRA

mu_star restricted_to M
 -> MEASURE

OUTER_MEASURE
 -> generates(MEASURE_THEORETIC_STRUCTURE)

# ---------- INDICATOR FUNCTION ----------
indicator_A(x)
 :=
1 if x in A
0 if x notin A

A measurable

iff

indicator_A
 -> MEASURABLE_FUNCTION

INDICATOR_FUNCTION
 -> function_representation_of(MEASURABLE_SET)

# ---------- SIMPLE FUNCTIONS ----------
phi(x)
 :=
sum_from_k_1_to_n(
 a_k
 *
 indicator_(A_k)(x)
)

if
 A_k measurable

then

phi
 -> SIMPLE_MEASURABLE_FUNCTION

MEASURABLE_SETS
 -> building_blocks_of(SIMPLE_FUNCTIONS)

# ---------- PREIMAGE ----------
f:
 (X,Sigma)
 ->
(Y,Tau)

f measurable

iff

for_all(B in Tau):

f_inverse(B)
 in Sigma

MEASURABLE_SET_IN_TARGET
 -> has(MEASURABLE_PREIMAGE)

MEASURABLE_FUNCTION
 -> transports(MEASURABILITY_BACKWARD_BY_PREIMAGE)

# ---------- LEVEL SETS ----------
for measurable real-valued f:

{x : f(x)>a}
 -> measurable

{x : f(x)>=a}
 -> measurable

{x : f(x)<a}
 -> measurable

{x : f(x)<=a}
 -> measurable

LEVEL_SETS
 -> measurable

# ---------- CONTINUOUS FUNCTIONS ----------
if
 X,Y topological spaces
with Borel sigma algebras

and
 f:X->Y continuous

then

f
 -> BOREL_MEASURABLE

OPEN_SET_PREIMAGE
 -> OPEN
 -> BOREL

CONTINUITY
 -> implies(BOREL_MEASURABILITY)

# ---------- PRODUCT SET ----------
A in Sigma
B in Tau

then

A x B
 -> measurable_in(Sigma tensor Tau)

MEASURABLE_RECTANGLE
 -> generator_of(PRODUCT_SIGMA_ALGEBRA)

PRODUCT_MEASURABLE_SET
 -> may_be_more_general_than(SINGLE_RECTANGLE)

# ---------- TRACE MEASURABILITY ----------
for measurable or arbitrary subset E with trace structure:

Sigma_E
 :=
{
 A intersection E :
 A in Sigma
}

B subset_of E

B
 -> measurable_in(E)

iff

B
 =
A intersection E
for some A in Sigma

TRACE_SIGMA_ALGEBRA
 -> inherited_measurability

# ---------- COMPLETION ----------
given
 (X,Sigma,mu)

completed sigma algebra:

Sigma_bar
 -> contains(Sigma)

and
 -> contains(SUBSETS_OF_mu_NULL_SETS)

A
 -> measurable_in_completion
 if
 A differs_from_some_B_in_Sigma
 by subset_of_null_set

COMPLETION
 -> expands(MEASURABLE_SET_CLASS)

# ---------- MEASURABLE PARTITION ----------
{A_i}

-> MEASURABLE_PARTITION

if
 each A_i measurable
 pairwise_disjoint
 union_i A_i = X

MEASURABLE_PARTITION
 -> supports(SIMPLE_FUNCTIONS_AND_CONDITIONAL_STRUCTURES)

# ---------- MONOTONE SET SEQUENCES ----------
A_n increasing
 -> union_n A_n measurable

A_n decreasing
 -> intersection_n A_n measurable

MEASURE_CONTINUITY
 -> applies_to(MONOTONE_MEASURABLE_SET_SEQUENCES)

# ---------- PROBABILITY EVENT ----------
probability space:

(Omega,F,P)

EVENT
 := A in F

EVENT
 -> MEASURABLE_SET

P(A)
 -> defined_for(EVENT)

PROBABILITY_EVENT
 ~= measurable_subset_of(SAMPLE_SPACE)

# ---------- NONMEASURABLE SET ----------
A subset_of X

A notin Sigma

-> NONMEASURABLE_RELATIVE_TO(Sigma)

NONMEASURABILITY
 -> does_not_mean(SET_DOES_NOT_EXIST)

it means

A
 -> outside(SELECTED_MEASURABLE_STRUCTURE)

# ---------- APPLICATIONS ----------
MEASURABLE_SET
 -> applied_to(MEASURE_ASSIGNMENT)
 -> applied_to(LEBESGUE_INTEGRATION)
 -> applied_to(PROBABILITY_EVENTS)
 -> applied_to(MEASURABLE_FUNCTIONS)
 -> applied_to(CONVERGENCE_THEOREMS)
 -> applied_to(PRODUCT_MEASURES)
 -> applied_to(STOCHASTIC_ANALYSIS)

# ---------- RELATION GRAPH ----------
UNDERLYING_SET
 -> equipped_with(SIGMA_ALGEBRA)

SIGMA_ALGEBRA
 -> contains(MEASURABLE_SETS)

MEASURABLE_SET
 -> receives(MEASURE)

MEASURABLE_SET
 -> generates(INDICATOR_FUNCTION)

MEASURABLE_SETS
 -> generate(SIMPLE_FUNCTIONS)

NULL_SET
 -> special_case_of(MEASURABLE_SET)

BOREL_SET
 -> measurable_under(BOREL_SIGMA_ALGEBRA)

COMPLETION
 -> enlarges(MEASURABLE_SET_CLASS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "measurable set" ~= "subset belonging to a specified sigma algebra",
 "Borel set" ~= "set belonging to the sigma algebra generated by open sets",
 "Lebesgue measurable set" ~= "set belonging to the completed Lebesgue measurable structure",
 "null set" ~= "measurable set of measure zero",
 "Caratheodory measurable set" ~= "set splitting outer measure additively for every test set",
 "event" ~= "measurable subset of a probability sample space"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 measurability_is_relative_to_sigma_algebra: true,
 not_every_subset_is_measurable_in_general: true,
 countable_unions_and_intersections_of_measurable_sets_are_measurable: true,
 Borel_and_Lebesgue_measurability_are_distinct: true,
 subsets_of_null_sets_are_measurable_only_automatically_in_complete_measure_spaces: true,
 outer_measure_can_be_defined_for_nonmeasurable_sets_but_is_not_same_as_measure: true,
 probability_events_must_belong_to_event_sigma_algebra: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 sigma_membership_relation: 0.99,
 closure_relation: 0.99,
 null_set_relation: 0.99,
 Borel_Lebesgue_relation: 0.99,
 Caratheodory_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(underlying_set_and_sigma_algebra)
 -> identify(candidate_subset)
 -> test(membership_in_sigma_algebra)
 -> use(closure_properties_if_set_built_from_measurable_sets)
 -> classify(Borel_Lebesgue_null_product_or_probability_event_if_relevant)
 -> analyse(Caratheodory_condition_if_outer_measure_context)
 -> determine(measure_value_if_measure_defined)
 -> connect_to(indicator_or_measurable_function_if_requested)
 -> validate(completeness_and_null_subset_conditions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed measurable-set semantic graph",
 primary_entity: MEASURABLE_SET,
 preserve: [
  underlying_set,
  sigma_algebra,
  subset,
  measure,
  completeness,
  topology,
  outer_measure,
  product_structure
 ],
 infer_through: [
  complements,
  countable_unions,
  countable_intersections,
  null_sets,
  Borel_sets,
  Lebesgue_measurable_sets,
  Caratheodory_measurability,
  indicator_functions
 ],
 avoid: [
  calling_set_measurable_without specifying structure,
  assuming_all_subsets_are measurable,
  confusing_outer_measure_with measure,
  assuming_every_subset_of_null_set_is measurable_in incomplete space,
  treating_Borel_and_Lebesgue_sigma_algebras_as identical,
  assigning_probability_to non-event
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(measurable_set_context)
 -> compare(existing_relations)
 -> preserve(space_sigma_subset_measure_and_completion_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + MEASURABLE_SET
 + UNDERLYING_SET
 + SIGMA_ALGEBRA
 + MEASURE
 + NULL_SET
 + BOREL_SET
 + LEBESGUE_MEASURABLE_SET
 + CARATHEODORY_MEASURABLE_SET
 + INDICATOR_FUNCTION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> MEASURABLE_SET
 -> SPACE_SIGMA_IDENTIFICATION
 -> SUBSET_IDENTIFICATION
 -> SIGMA_MEMBERSHIP_ANALYSIS
 -> CLOSURE_ANALYSIS
 -> SET_TYPE_CLASSIFICATION
 -> CARATHEODORY_ANALYSIS
 -> MEASURE_ANALYSIS
 -> FUNCTION_RELATION_ANALYSIS
 -> COMPLETENESS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW