Measure Space Ontology

flow "MeasureSpaceOntology" v1.0

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

MEASURE_SPACE := {
 type: MeasureTheoreticStructure,
 domain: MeasureTheory,
 purpose: "Represent an underlying set equipped with a sigma algebra of measurable subsets and a measure assigning size to those subsets."
}

CONCEPTS := {
 UNDERLYING_SET,
 SIGMA_ALGEBRA,
 MEASURE,
 MEASURE_SPACE,
 MEASURABLE_SET,
 MEASURABLE_FUNCTION,
 NULL_SET,
 COMPLETE_MEASURE_SPACE,
 FINITE_MEASURE_SPACE,
 SIGMA_FINITE_MEASURE_SPACE,
 PROBABILITY_SPACE,
 PRODUCT_MEASURE_SPACE,
 LEBESGUE_MEASURE_SPACE,
 INTEGRATION,
 ALMOST_EVERYWHERE
}

# ---------- DEFINITION ----------
MEASURE_SPACE
 := (X,Sigma,mu)

X
 -> UNDERLYING_SET

Sigma
 -> SIGMA_ALGEBRA_ON(X)

mu
 -> MEASURE_ON(Sigma)

MEASURE_SPACE
 -> combines(SET_STRUCTURE, MEASURABILITY, SIZE)

# ---------- SIGMA ALGEBRA ----------
Sigma subset_of POWER_SET(X)

X in Sigma

A in Sigma
 -> X\A in Sigma

A_n in Sigma
 -> union_n(A_n) in Sigma

SIGMA_ALGEBRA
 -> determines(MEASURABLE_SETS)

# ---------- MEASURE ----------
mu:
 Sigma -> [0,infinity]

mu(empty_set)=0

PAIRWISE_DISJOINT(A_n)
 -> mu(union_n A_n)
    = sum_n(mu(A_n))

MEASURE
 -> COUNTABLY_ADDITIVE

# ---------- MEASURABLE SET ----------
A subset_of X

A
 -> MEASURABLE_SET

iff

A in Sigma

MEASURABLE_SET
 -> eligible_for(MEASURE_ASSIGNMENT)

# ---------- NULL SET ----------
N in Sigma

N
 -> NULL_SET

iff

mu(N)=0

NULL_SET
 -> negligible_in(MEASURE_THEORETIC_SENSE)

# ---------- ALMOST EVERYWHERE ----------
property P

P
 -> holds(ALMOST_EVERYWHERE)

iff

mu(
 {x in X : P(x) fails}
)
=0

MEASURE_SPACE
 -> defines(ALMOST_EVERYWHERE_CONTEXT)

# ---------- COMPLETE MEASURE SPACE ----------
(X,Sigma,mu)
 -> COMPLETE

if

N in Sigma
and
mu(N)=0
and
A subset_of N

implies

A in Sigma

COMPLETE_MEASURE_SPACE
 -> contains(ALL_SUBSETS_OF_NULL_SETS)

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

COMPLETION
 -> enlarged_sigma_algebra(Sigma_bar)

such_that

Sigma subset_of Sigma_bar

and

all subsets of mu-null sets
 -> belong_to(Sigma_bar)

mu
 -> extended_to(COMPLETED_MEASURE)

# ---------- FINITE MEASURE SPACE ----------
MEASURE_SPACE
 -> FINITE

if

mu(X)
 < infinity

FINITE_MEASURE_SPACE
 -> total_measure_finite

# ---------- PROBABILITY SPACE ----------
if

mu(X)=1

then

(X,Sigma,mu)
 -> PROBABILITY_SPACE

PROBABILITY_SPACE
 -> NORMALISED_FINITE_MEASURE_SPACE

# ---------- SIGMA FINITE ----------
MEASURE_SPACE
 -> SIGMA_FINITE

if

exists
 E_1,E_2,...

such_that

X
 =
union_n(E_n)

and

mu(E_n)
 < infinity

for_all n

SIGMA_FINITE
 -> important_for(
 FUBINI,
 PRODUCT_MEASURE,
 RADON_NIKODYM
)

# ---------- SEMIFINITE ----------
MEASURE
 -> SEMIFINITE

if whenever

mu(A)=infinity

there exists
 B subset_of A

with

0 < mu(B) < infinity

SEMIFINITENESS
 -> intermediate_measure_property

# ---------- ATOMIC MEASURE SPACE ----------
ATOM
 -> measurable A with mu(A)>0

such_that every measurable
 B subset_of A

has

mu(B)=0
or
mu(B)=mu(A)

ATOMIC_MEASURE_SPACE
 -> contains(MEASURE_ATOMS)

# ---------- NONATOMIC SPACE ----------
NONATOMIC_MEASURE_SPACE
 -> has(NO_POSITIVE_MEASURE_ATOMS)

LEBESGUE_MEASURE_ON_R
 -> NONATOMIC

COUNTING_MEASURE
 -> typically(ATOMIC)

# ---------- LEBESGUE MEASURE SPACE ----------
(R^n, L^n, lambda_n)

where

L^n
 -> LEBESGUE_SIGMA_ALGEBRA

lambda_n
 -> LEBESGUE_MEASURE

LEBESGUE_MEASURE_SPACE
 -> standard_Euclidean_measure_space

# ---------- COUNTING MEASURE SPACE ----------
(X,POWER_SET(X),#)

where

#(A)
 -> number_of_elements_of(A)

COUNTING_MEASURE_SPACE
 -> discrete_measure_space

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

(X,Sigma,delta_x0)

delta_x0(A)
 :=
1 if x0 in A
0 otherwise

DIRAC_MEASURE_SPACE
 -> measure_concentrated_at(SINGLE_POINT)

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

f
 -> MEASURABLE

iff

for_all(B in Tau):

f_inverse(B)
 in Sigma

MEASURE_SPACE
 -> domain_for(MEASURABLE_FUNCTIONS)

# ---------- INTEGRATION ----------
for measurable f:

integral_X(
 f dmu
)

-> LEBESGUE_INTEGRAL_WITH_RESPECT_TO(mu)

MEASURE_SPACE
 -> supplies(DOMAIN_AND_MEASURE_FOR_INTEGRATION)

# ---------- Lp SPACES ----------
for
 1<=p<infinity

Lp(X,Sigma,mu)
 := {
 measurable f :
 integral_X(
  |f|^p dmu
 ) < infinity
 }
 modulo(ALMOST_EVERYWHERE_EQUALITY)

MEASURE_SPACE
 -> generates(FUNCTION_SPACES)

# ---------- CONVERGENCE IN MEASURE ----------
f_n
 -> f
 in measure

if for every epsilon>0:

mu(
 {x : |f_n(x)-f(x)|>epsilon}
)
 -> 0

CONVERGENCE_IN_MEASURE
 -> depends_on(MEASURE_SPACE)

# ---------- ALMOST EVERYWHERE CONVERGENCE ----------
f_n
 -> f
 almost_everywhere

if

mu(
 {x : f_n(x) does_not_converge_to f(x)}
)
=0

MEASURE_SPACE
 -> determines(EXCEPTION_SET_SIZE)

# ---------- PRODUCT MEASURE SPACE ----------
(X,Sigma,mu)
and
(Y,Tau,nu)

->

(X x Y,
 Sigma tensor Tau,
 mu tensor nu)

PRODUCT_MEASURE_SPACE
 -> supports(MULTIVARIABLE_INTEGRATION)

# ---------- PRODUCT SIGMA ALGEBRA ----------
Sigma tensor Tau
 :=
sigma(
 {
  A x B :
  A in Sigma,
  B in Tau
 }
)

MEASURABLE_RECTANGLES
 -> generate(PRODUCT_STRUCTURE)

# ---------- PRODUCT MEASURE ----------
(mu tensor nu)(A x B)
 =
mu(A)*nu(B)

under standard product measure construction

PRODUCT_MEASURE
 -> extends(RECTANGLE_MEASURE_MULTIPLICATIVELY)

# ---------- FUBINI / TONELLI ----------
PRODUCT_MEASURE_SPACE
 -> context_for(FUBINI_THEOREM)
 -> context_for(TONELLI_THEOREM)

TONELLI
 -> nonnegative_measurable_functions

FUBINI
 -> integrable_functions

# ---------- RESTRICTION ----------
for measurable
 E in Sigma

restricted space:

(E,
 Sigma_E,
 mu_E)

where

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

mu_E(B)
 :=
mu(B)

RESTRICTED_MEASURE_SPACE
 -> localises(MEASURE_STRUCTURE)

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

T:
 (X,Sigma)
 ->
(Y,Tau)

pushforward:

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

(Y,Tau,T_sharp(mu))
 -> PUSHFORWARD_MEASURE_SPACE

MEASURABLE_MAP
 -> transports(MEASURE)

# ---------- EQUIVALENT MEASURES ----------
mu
and
nu

-> EQUIVALENT

if

mu << nu
and
nu << mu

EQUIVALENT_MEASURES
 -> share(NULL_SETS)

MEASURE_SPACES_WITH_EQUIVALENT_MEASURES
 -> share(ALMOST_EVERYWHERE_STRUCTURE)

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

if

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

ABSOLUTE_CONTINUITY
 -> relation_between(MEASURES_ON_SAME_MEASURABLE_SPACE)

# ---------- RADON-NIKODYM ----------
if

nu << mu

and suitable hypotheses

then

dnu/dmu
 -> MEASURABLE_DENSITY

nu(A)
 =
integral_A(
 dnu/dmu
 dmu
)

MEASURE_SPACE
 -> context_for(CHANGE_OF_MEASURE)

# ---------- REGULAR MEASURE SPACE ----------
in topological setting:

measure
 -> REGULAR

if measurable sets can be approximated
from outside by open sets
and from inside by compact sets
under appropriate hypotheses

REGULARITY
 -> links(TOPOLOGY_AND_MEASURE)

# ---------- APPLICATIONS ----------
MEASURE_SPACE
 -> applied_to(LEBESGUE_INTEGRATION)
 -> applied_to(PROBABILITY)
 -> applied_to(FUNCTIONAL_ANALYSIS)
 -> applied_to(FOURIER_ANALYSIS)
 -> applied_to(ERGODIC_THEORY)
 -> applied_to(STOCHASTIC_PROCESSES)
 -> applied_to(PDEs)
 -> applied_to(MATHEMATICAL_PHYSICS)

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

SIGMA_ALGEBRA
 -> supports(MEASURE)

MEASURE
 -> assigns(SIZE_TO_MEASURABLE_SETS)

MEASURE_SPACE
 -> defines(NULL_SETS)

NULL_SETS
 -> define(ALMOST_EVERYWHERE)

MEASURE_SPACE
 -> supports(INTEGRATION)

MEASURE_SPACE
 -> generates(Lp_SPACES)

PRODUCT
 -> combines(MEASURE_SPACES)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "measure space" ~= "triple consisting of a set, sigma algebra and measure",
 "complete measure space" ~= "measure space containing every subset of each measurable null set",
 "finite measure space" ~= "measure space whose whole underlying set has finite measure",
 "sigma-finite measure space" ~= "measure space coverable by countably many finite-measure measurable sets",
 "probability space" ~= "measure space with total measure one",
 "product measure space" ~= "measure space formed from measurable product structure and product measure"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 measure_space_requires_set_sigma_algebra_and_measure: true,
 measure_must_be_defined_on_selected_sigma_algebra: true,
 completeness_is_additional_property_not_automatic: true,
 finite_and_sigma_finite_are_distinct_properties: true,
 probability_space_requires_total_measure_one: true,
 product_measure_construction_requires_measure_theoretic_hypotheses: true,
 convergence_and_almost_everywhere_statements_depend_on_measure: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 sigma_measure_relation: 0.99,
 completeness_relation: 0.99,
 finiteness_relation: 0.99,
 integration_relation: 0.99,
 product_space_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(underlying_set_sigma_algebra_and_measure)
 -> verify(measure_axioms_and_domain)
 -> classify(complete_finite_sigma_finite_atomic_or_probability_structure)
 -> identify(measurable_sets_and_null_sets)
 -> determine(almost_everywhere_and_convergence_context)
 -> analyse(measurable_functions_and_integrability)
 -> construct(product_restriction_or_pushforward_space_if_requested)
 -> analyse(absolute_continuity_or_change_of_measure_if_present)
 -> validate(finiteness_completeness_and_product_hypotheses)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed measure-space semantic graph",
 primary_entity: MEASURE_SPACE,
 preserve: [
  underlying_set,
  sigma_algebra,
  measure,
  measurable_sets,
  null_sets,
  completeness,
  finiteness_type,
  topology
 ],
 infer_through: [
  measurable_functions,
  Lebesgue_integrals,
  Lp_spaces,
  convergence_in_measure,
  almost_everywhere_convergence,
  product_measure_spaces,
  restrictions,
  pushforwards
 ],
 avoid: [
  omitting_sigma_algebra_from measure space,
  defining_measure_on arbitrary nonmeasurable subsets,
  assuming_measure_space_is complete,
  confusing_finite_with sigma_finite,
  treating_probability_space_as arbitrary measure space,
  ignoring_measure_when interpreting almost_everywhere
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + MEASURE_SPACE
 + UNDERLYING_SET
 + SIGMA_ALGEBRA
 + MEASURE
 + NULL_SET
 + COMPLETE_MEASURE_SPACE
 + FINITE_MEASURE_SPACE
 + SIGMA_FINITE_MEASURE_SPACE
 + PRODUCT_MEASURE_SPACE
 + INTEGRATION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> MEASURE_SPACE
 -> SPACE_SIGMA_MEASURE_IDENTIFICATION
 -> MEASURE_VALIDATION
 -> SPACE_CLASSIFICATION
 -> NULL_SET_ANALYSIS
 -> CONVERGENCE_CONTEXT_ANALYSIS
 -> MEASURABLE_FUNCTION_ANALYSIS
 -> INTEGRATION_ANALYSIS
 -> PRODUCT_RESTRICTION_PUSHFORWARD_ANALYSIS
 -> CHANGE_OF_MEASURE_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW