Measure Ontology

flow "MeasureOntology" v1.0

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

MEASURE := {
 type: CountablyAdditiveSetFunction,
 domain: MeasureTheory,
 purpose: "Assign a nonnegative extended-real size to measurable sets while preserving countable additivity."
}

CONCEPTS := {
 MEASURE,
 SIGMA_ALGEBRA,
 MEASURABLE_SET,
 MEASURE_SPACE,
 COUNTABLE_ADDITIVITY,
 NULL_SET,
 FINITE_MEASURE,
 SIGMA_FINITE_MEASURE,
 PROBABILITY_MEASURE,
 LEBESGUE_MEASURE,
 COUNTING_MEASURE,
 DIRAC_MEASURE,
 PRODUCT_MEASURE,
 SIGNED_MEASURE
}

# ---------- DEFINITION ----------
let
 X -> UNDERLYING_SET
 Sigma -> SIGMA_ALGEBRA_ON(X)

mu:
 Sigma -> [0,infinity]

MEASURE(mu)
 iff

mu(empty_set)=0

AND

for every pairwise_disjoint sequence A_n in Sigma:

mu(union_n(A_n))
 =
sum_n(mu(A_n))

COUNTABLE_ADDITIVITY
 -> defining_axiom_of(MEASURE)

# ---------- MEASURE SPACE ----------
MEASURE_SPACE
 := (X,Sigma,mu)

MEASURE
 -> acts_on(MEASURABLE_SETS)

A
 -> MEASURABLE
 iff(A in Sigma)

# ---------- NONNEGATIVITY ----------
for_all(A in Sigma):

mu(A)
 >= 0

MEASURE
 -> NONNEGATIVE

# ---------- MONOTONICITY ----------
if
 A subset_of B

then

mu(A)
 <=
mu(B)

MONOTONICITY
 -> derived_from(COUNTABLE_ADDITIVITY)

# ---------- SUBADDITIVITY ----------
for measurable A_n:

mu(union_n(A_n))
 <=
sum_n(mu(A_n))

COUNTABLE_SUBADDITIVITY
 -> derived_measure_property

# ---------- FINITE ADDITIVITY ----------
if
 A intersection B
 = empty_set

then

mu(A union B)
 =
mu(A)+mu(B)

COUNTABLE_ADDITIVITY
 -> implies(FINITE_ADDITIVITY)

# ---------- DIFFERENCE ----------
if
 A subset_of B

and
mu(A)<infinity

then

mu(B\A)
 =
mu(B)-mu(A)

FINITE_SUBMEASURE
 -> enables(DIFFERENCE_FORMULA)

# ---------- CONTINUITY FROM BELOW ----------
A_1 subset_of A_2 subset_of ...

then

mu(union_n A_n)
 =
lim_n mu(A_n)

MEASURE
 -> CONTINUOUS_FROM_BELOW

# ---------- CONTINUITY FROM ABOVE ----------
A_1 superset_of A_2 superset_of ...

AND
mu(A_1)<infinity

then

mu(intersection_n A_n)
 =
lim_n mu(A_n)

CONTINUITY_FROM_ABOVE
 -> requires(FINITE_INITIAL_MEASURE_IN_STANDARD_FORM)

# ---------- NULL SET ----------
N
 -> NULL_SET

iff

mu(N)=0

NULL_SET
 -> negligible_under(MEASURE)

PROPERTY
 -> holds(ALMOST_EVERYWHERE)
 if
 failure_set -> NULL_SET

# ---------- COMPLETE MEASURE ----------
mu
 -> COMPLETE

if every subset of each measurable null set
 -> measurable

COMPLETION
 -> extends(SIGMA_ALGEBRA)

# ---------- FINITE MEASURE ----------
mu
 -> FINITE_MEASURE

if

mu(X)<infinity

FINITE_MEASURE_SPACE
 -> total_measure_finite

# ---------- PROBABILITY MEASURE ----------
P
 -> PROBABILITY_MEASURE

if

P(X)=1

PROBABILITY_MEASURE
 -> finite_measure
 -> normalised_measure

# ---------- SIGMA FINITE ----------
mu
 -> SIGMA_FINITE

if

X
 =
union_n(E_n)

with

mu(E_n)<infinity

SIGMA_FINITE
 -> important_hypothesis_for(
    PRODUCT_MEASURES,
    RADON_NIKODYM,
    FUBINI_THEORY
 )

# ---------- LOCALLY FINITE ----------
mu
 -> LOCALLY_FINITE

if suitable neighbourhoods_or_compact_sets
 -> have_finite_measure
 under(topological_context)

LOCAL_FINITENESS
 -> important_for(RADON_MEASURES)

# ---------- LEBESGUE MEASURE ----------
lambda
 -> LEBESGUE_MEASURE_ON(R^n)

in R:

lambda([a,b])
 =
b-a

LEBESGUE_MEASURE
 -> TRANSLATION_INVARIANT

lambda(A+x)
 =
lambda(A)

for measurable A

# ---------- COUNTING MEASURE ----------
COUNTING_MEASURE:

mu(A)
 :=
number_of_elements_in(A)

with
 infinity_if_A_infinite

COUNTING_MEASURE
 -> measure_on(POWER_SET_FOR_COUNTABLE_OR_GENERAL_SET_CONTEXT)

# ---------- DIRAC MEASURE ----------
for fixed point x0:

delta_x0(A)
 :=
1
 if x0 in A

0
 otherwise

DIRAC_MEASURE
 -> concentrated_at(SINGLE_POINT)

integral(
 f d delta_x0
)
 =
f(x0)

for suitable measurable f

# ---------- WEIGHTED MEASURE ----------
given base measure mu
and measurable density w>=0:

nu(A)
 :=
integral_A(
 w dmu
)

nu
 -> MEASURE

w
 -> DENSITY_WITH_RESPECT_TO(mu)

# ---------- ABSOLUTE CONTINUITY ----------
nu << mu

if

mu(A)=0
 -> nu(A)=0

DENSITY_DEFINED_MEASURE
 -> ABSOLUTELY_CONTINUOUS_WITH_RESPECT_TO_BASE_MEASURE

# ---------- PRODUCT MEASURE ----------
mu on X
nu on Y

PRODUCT_MEASURE
 := mu tensor nu

on measurable rectangles:

(mu tensor nu)(A x B)
 =
mu(A)*nu(B)

PRODUCT_MEASURE
 -> supports(MULTIDIMENSIONAL_INTEGRATION)

# ---------- PUSHFORWARD MEASURE ----------
measurable map

T:X -> Y

pushforward:

T_sharp mu(B)
 :=
mu(
 T_inverse(B)
)

PUSHFORWARD_MEASURE
 -> transports(MEASURE_THROUGH_MEASURABLE_MAP)

# ---------- RESTRICTION ----------
for measurable E:

mu_restricted_to_E(A)
 :=
mu(A intersection E)

RESTRICTED_MEASURE
 -> localises(MEASURE_TO_SUBSET)

# ---------- SCALING ----------
for c>=0:

(c*mu)(A)
 :=
c*mu(A)

SCALAR_MULTIPLE_OF_MEASURE
 -> MEASURE

# ---------- SUM OF MEASURES ----------
if
 mu_n -> measures

and
 a_n>=0

then

nu(A)
 :=
sum_n(
 a_n*mu_n(A)
)

when well-defined

nu
 -> MEASURE

MEASURES
 -> closed_under(NONNEGATIVE_LINEAR_COMBINATIONS)

# ---------- SIGNED MEASURE DISTINCTION ----------
SIGNED_MEASURE
 -> may_take(NEGATIVE_VALUES)

MEASURE
 -> NONNEGATIVE

SIGNED_MEASURE
 -/> ordinary_positive_measure

JORDAN_DECOMPOSITION
 -> expresses(SIGNED_MEASURE_AS_DIFFERENCE_OF_POSITIVE_MEASURES)

# ---------- OUTER MEASURE DISTINCTION ----------
OUTER_MEASURE
 -> defined_on(ALL_SUBSETS)
 -> COUNTABLY_SUBADDITIVE

MEASURE
 -> defined_on(SIGMA_ALGEBRA)
 -> COUNTABLY_ADDITIVE

OUTER_MEASURE
 -/> automatically(MEASURE_ON_ALL_SUBSETS)

CARATHEODORY
 -> identifies(MEASURABLE_SUBSETS_FOR_OUTER_MEASURE)

# ---------- INTEGRATION ----------
MEASURE
 -> determines(INTEGRAL)

integral_X(
 f dmu
)

-> LEBESGUE_INTEGRAL_WITH_RESPECT_TO(mu)

CHANGE_OF_MEASURE
 -> changes(INTEGRAL_AND_NULL_SET_STRUCTURE)

# ---------- CONVERGENCE ----------
measure mu
 -> defines(
    ALMOST_EVERYWHERE_CONVERGENCE,
    CONVERGENCE_IN_MEASURE,
    Lp_CONVERGENCE
 )

MEASURE
 -> foundational_to(MEASURE_THEORETIC_CONVERGENCE)

# ---------- APPLICATIONS ----------
MEASURE
 -> applied_to(LEBESGUE_INTEGRATION)
 -> applied_to(PROBABILITY)
 -> applied_to(FUNCTIONAL_ANALYSIS)
 -> applied_to(FOURIER_ANALYSIS)
 -> applied_to(ERGODIC_THEORY)
 -> applied_to(GEOMETRIC_MEASURE_THEORY)
 -> applied_to(STOCHASTIC_ANALYSIS)

# ---------- RELATION GRAPH ----------
SIGMA_ALGEBRA
 -> supplies(DOMAIN_OF_MEASURE)

MEASURE
 -> assigns(SIZE_TO_MEASURABLE_SET)

MEASURE
 -> defines(NULL_SETS)

MEASURE
 -> supports(INTEGRATION)

MEASURE
 -> supports(ALMOST_EVERYWHERE_PROPERTIES)

SIGMA_FINITE_MEASURE
 -> enables(MAJOR_MEASURE_THEOREMS)

DENSITY
 -> generates(ABSOLUTELY_CONTINUOUS_MEASURE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "measure" ~= "countably additive nonnegative set function on a sigma algebra",
 "finite measure" ~= "measure whose total space has finite measure",
 "probability measure" ~= "measure normalised to total mass one",
 "sigma-finite measure" ~= "measure whose space is a countable union of finite-measure sets",
 "Dirac measure" ~= "unit point mass concentrated at one point",
 "pushforward measure" ~= "measure transported through a measurable mapping"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 measure_is_nonnegative: true,
 countable_additivity_is_required: true,
 measure_domain_is_sigma_algebra_in_general: true,
 continuity_from_above_standard_form_requires_finite_initial_measure: true,
 probability_measure_has_total_mass_one: true,
 signed_measure_is_distinct_from_positive_measure: true,
 outer_measure_is_not_automatically_measure_on_all_subsets: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 countable_additivity_relation: 0.99,
 continuity_relation: 0.99,
 null_set_relation: 0.99,
 measure_type_relation: 0.99,
 integration_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(underlying_set_and_sigma_algebra)
 -> identify(candidate_measure)
 -> verify(nonnegativity_and_empty_set_axiom)
 -> verify(countable_additivity)
 -> classify(finite_probability_sigma_finite_or_other_type)
 -> identify(null_sets_and_completion_status)
 -> analyse(continuity_subadditivity_and_restriction_properties)
 -> analyse(product_pushforward_or_density_construction_if_relevant)
 -> connect_to(integration_or_convergence_if_requested)
 -> validate(measurability_and_finiteness_hypotheses)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed measure semantic graph",
 primary_entity: MEASURE,
 preserve: [
  underlying_set,
  sigma_algebra,
  measure,
  measurable_sets,
  total_measure,
  null_sets,
  finiteness_type,
  completeness
 ],
 infer_through: [
  monotonicity,
  subadditivity,
  continuity_properties,
  Lebesgue_measure,
  counting_measure,
  Dirac_measure,
  product_measures,
  pushforward_measures,
  density_measures
 ],
 avoid: [
  treating_finitely_additive_function_as measure_without_countable_additivity,
  assigning_measure_outside_sigma_algebra_without_outer_measure_context,
  confusing_signed_measure_with measure,
  omitting_finiteness_condition_for_continuity_from_above,
  confusing_probability_measure_with arbitrary finite measure,
  treating_outer_measure_as ordinary measure_without_Caratheodory_restriction
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + MEASURE
 + SIGMA_ALGEBRA
 + COUNTABLE_ADDITIVITY
 + NULL_SET
 + FINITE_MEASURE
 + SIGMA_FINITE_MEASURE
 + PROBABILITY_MEASURE
 + LEBESGUE_MEASURE
 + PRODUCT_MEASURE
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> MEASURE
 -> SPACE_SIGMA_IDENTIFICATION
 -> AXIOM_VALIDATION
 -> ADDITIVITY_ANALYSIS
 -> MEASURE_TYPE_CLASSIFICATION
 -> NULL_SET_COMPLETION_ANALYSIS
 -> CONTINUITY_PROPERTY_ANALYSIS
 -> MEASURE_CONSTRUCTION_ANALYSIS
 -> INTEGRATION_CONVERGENCE_RELATION
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW