Measure Theory Ontology

flow "MeasureTheoryOntology" v1.0

# ============================================================
# MEASURE THEORY ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

MEASURE_THEORY := {
 type: MathematicalAnalysisFramework,
 purpose: "Formalise measurable sets, measures, integration, almost-everywhere properties and convergence on measure spaces."
}

CONCEPTS := {
 SIGMA_ALGEBRA,
 MEASURABLE_SPACE,
 MEASURE,
 MEASURE_SPACE,
 MEASURABLE_SET,
 MEASURABLE_FUNCTION,
 NULL_SET,
 ALMOST_EVERYWHERE,
 OUTER_MEASURE,
 LEBESGUE_MEASURE,
 LEBESGUE_INTEGRAL,
 PRODUCT_MEASURE,
 SIGNED_MEASURE,
 ABSOLUTE_CONTINUITY,
 RADON_NIKODYM_DERIVATIVE
}

# ---------- SIGMA ALGEBRA ----------
SIGMA_ALGEBRA := Sigma

Sigma subset_of POWER_SET(X)

X in Sigma

A in Sigma
 -> X\A in Sigma

A_1,A_2,... in Sigma
 -> union_n(A_n) in Sigma

SIGMA_ALGEBRA
 -> closed_under(COUNTABLE_COMPLEMENTS_AND_UNIONS)

# ---------- MEASURABLE SPACE ----------
MEASURABLE_SPACE
 := (X,Sigma)

A
 -> MEASURABLE_SET
 iff(A in Sigma)

# ---------- MEASURE ----------
MEASURE := mu

mu:
 Sigma -> [0,infinity]

mu(empty_set)=0

if
 A_n pairwise_disjoint

then

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

COUNTABLE_ADDITIVITY
 -> defining_property_of(MEASURE)

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

MEASURE_SPACE
 -> supports(MEASURABILITY_AND_INTEGRATION)

# ---------- BASIC PROPERTIES ----------
A subset_of B
 -> mu(A) <= mu(B)

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

MEASURE
 -> MONOTONE
 -> COUNTABLY_SUBADDITIVE

# ---------- CONTINUITY ----------
if
 A_n increasing

then

mu(union_n A_n)
 =
lim_n mu(A_n)

if
 A_n decreasing
 AND mu(A_1)<infinity

then

mu(intersection_n A_n)
 =
lim_n mu(A_n)

# ---------- NULL SET ----------
N
 -> NULL_SET
 iff(mu(N)=0)

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

ALMOST_EVERYWHERE
 -> fundamental_equivalence_context

# ---------- COMPLETENESS ----------
MEASURE_SPACE
 -> COMPLETE

if
 every subset of measurable null set
 -> measurable

COMPLETION
 -> extends(SIGMA_ALGEBRA_BY_NULL_SUBSETS)

# ---------- BOREL STRUCTURE ----------
BOREL_SIGMA_ALGEBRA
 := sigma(OPEN_SETS)

BOREL_SET
 -> element_of(BOREL_SIGMA_ALGEBRA)

TOPOLOGY
 -> generates(BOREL_MEASURABLE_STRUCTURE)

# ---------- OUTER MEASURE ----------
OUTER_MEASURE := mu_star

mu_star:
 POWER_SET(X) -> [0,infinity]

mu_star(empty_set)=0

OUTER_MEASURE
 -> MONOTONE
 -> COUNTABLY_SUBADDITIVE

# ---------- CARATHEODORY ----------
A
 -> CARATHEODORY_MEASURABLE

if
 for_all(E subset_of X):

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

CARATHEODORY_CONSTRUCTION
 -> converts(OUTER_MEASURE_TO_MEASURE)

# ---------- LEBESGUE MEASURE ----------
LEBESGUE_MEASURE
 -> standard_measure_on(EUCLIDEAN_SPACE)

on R:

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

LEBESGUE_MEASURE
 -> TRANSLATION_INVARIANT
 -> COMPLETE_ON_LEBESGUE_SIGMA_ALGEBRA

# ---------- MEASURABLE FUNCTION ----------
f:X -> extended_real_numbers

f
 -> MEASURABLE

if
 preimage_of(Borel_set)
 -> measurable

for real-valued f equivalently:

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

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

A_k
 -> measurable

SIMPLE_FUNCTION
 -> finite_valued_measurable_function

SIMPLE_FUNCTION
 -> building_block_of(LEBESGUE_INTEGRAL)

# ---------- LEBESGUE INTEGRAL ----------
for
 f>=0 measurable

integral_X(f dmu)
 :=
sup_{0<=phi<=f, phi simple}(
 integral_X(phi dmu)
)

NONNEGATIVE_INTEGRAL
 -> extended_real_value

# ---------- INTEGRABLE FUNCTION ----------
f
 =
f_positive-f_negative

f_positive
 := max(f,0)

f_negative
 := max(-f,0)

if
 integral(f_positive dmu)
 +
 integral(f_negative dmu)
 < infinity

then

f
 -> L1_INTEGRABLE

integral(f dmu)
 =
integral(f_positive dmu)
-
integral(f_negative dmu)

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

Lp(mu)
 := {
 f :
 integral(|f|^p dmu)<infinity
 }
 modulo(ALMOST_EVERYWHERE_EQUALITY)

||f||_p
 :=
(
 integral(|f|^p dmu)
)^(1/p)

L_INFINITY
 -> ESSENTIAL_SUPREMUM_NORM

# ---------- MONOTONE CONVERGENCE ----------
if
 0<=f_n
 and
 f_n increasing_to f
 almost_everywhere

then

integral(f_n dmu)
 ->
integral(f dmu)

MONOTONE_CONVERGENCE_THEOREM
 -> exchanges(MONOTONE_LIMIT_AND_INTEGRAL)

# ---------- FATOU ----------
for
 f_n>=0 measurable

integral(
 liminf f_n
 dmu
)
<=
liminf(
 integral(f_n dmu)
)

FATOU_LEMMA
 -> lower_limit_integral_control

# ---------- DOMINATED CONVERGENCE ----------
if
 f_n -> f almost_everywhere

and
 |f_n| <= g

and
 g in L1

then

integral(f_n dmu)
 ->
integral(f dmu)

DOMINATED_CONVERGENCE_THEOREM
 -> limit_integral_interchange_under_domination

# ---------- CONVERGENCE MODES ----------
CONVERGENCE_MODES := {
 POINTWISE,
 ALMOST_EVERYWHERE,
 IN_MEASURE,
 IN_Lp,
 UNIFORM
}

CONVERGENCE_MODE
 -> must_be_specified

RELATIONS_BETWEEN_MODES
 -> require(MEASURE_SPACE_HYPOTHESES)

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

-> PRODUCT_SPACE

PRODUCT_SIGMA_ALGEBRA
 := Sigma tensor Tau

PRODUCT_MEASURE
 := mu tensor nu

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

# ---------- TONELLI ----------
if
 f>=0 measurable
 on product space

then

product_integral
 =
iterated_integrals

TONELLI_THEOREM
 -> supports(NONNEGATIVE_MEASURABLE_FUNCTIONS)

# ---------- FUBINI ----------
if
 f
 -> integrable_on_product_space

then

integral_product(f)
 =
iterated_integral_XY(f)
 =
iterated_integral_YX(f)

FUBINI_THEOREM
 -> changes(INTEGRATION_ORDER)

# ---------- SIGNED MEASURE ----------
SIGNED_MEASURE := nu

nu
 -> COUNTABLY_ADDITIVE
 -> may_take(POSITIVE_AND_NEGATIVE_VALUES)

JORDAN_DECOMPOSITION:

nu
 =
nu_positive
-
nu_negative

nu_positive
 and
nu_negative
 -> MUTUALLY_SINGULAR_POSITIVE_MEASURES

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

iff

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

ABSOLUTE_CONTINUITY
 -> preserves(MU_NULL_SETS)

# ---------- RADON-NIKODYM ----------
if
 nu << mu

and
 suitable_sigma_finiteness_conditions

then

exists
 f measurable

such_that

nu(A)
 =
integral_A(f dmu)

f
 :=
dnu/dmu

RADON_NIKODYM_DERIVATIVE
 -> density_of(nu_WITH_RESPECT_TO_mu)

# ---------- PROBABILITY ----------
PROBABILITY_SPACE
 := (Omega,F,P)

P
 -> MEASURE

P(Omega)=1

RANDOM_VARIABLE
 -> MEASURABLE_FUNCTION

EXPECTATION
 -> LEBESGUE_INTEGRAL

PROBABILITY_THEORY
 -> specialisation_of(MEASURE_THEORY)

# ---------- APPLICATIONS ----------
MEASURE_THEORY
 -> applied_to(REAL_ANALYSIS)
 -> applied_to(PROBABILITY)
 -> applied_to(FUNCTIONAL_ANALYSIS)
 -> applied_to(FOURIER_ANALYSIS)
 -> applied_to(PDEs)
 -> applied_to(ERGODIC_THEORY)
 -> applied_to(STOCHASTIC_ANALYSIS)
 -> applied_to(MATHEMATICAL_PHYSICS)

# ---------- RELATION GRAPH ----------
SIGMA_ALGEBRA
 -> defines(MEASURABLE_SETS)

MEASURE
 -> assigns(SIZE)

MEASURE_SPACE
 -> supports(MEASURABLE_FUNCTIONS)

MEASURABLE_FUNCTION
 -> supports(LEBESGUE_INTEGRATION)

NULL_SET
 -> defines(ALMOST_EVERYWHERE)

PRODUCT_MEASURE
 -> supports(FUBINI_AND_TONELLI)

ABSOLUTE_CONTINUITY
 -> supports(RADON_NIKODYM_DERIVATIVE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "measure theory" ~= "theory of measurable sets, measures and measure-based integration",
 "sigma algebra" ~= "set collection closed under complements and countable unions",
 "measure space" ~= "set equipped with a sigma algebra and measure",
 "measurable function" ~= "function compatible with measurable structures through measurable preimages",
 "almost everywhere" ~= "outside a set of measure zero",
 "Radon-Nikodym derivative" ~= "density of an absolutely continuous measure relative to another measure"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 measure_is_defined_on_measurable_sets_in_general: true,
 countable_additivity_is_required: true,
 almost_everywhere_is_not_pointwise_everywhere: true,
 convergence_modes_are_distinct: true,
 Fubini_requires_integrability: true,
 Tonelli_applies_to_nonnegative_measurable_functions: true,
 Radon_Nikodym_requires_absolute_continuity_and_measure_hypotheses: true,
 outer_measure_and_measure_are_distinct: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 sigma_measure_relation: 0.99,
 measurable_function_relation: 0.99,
 Lebesgue_integration_relation: 0.99,
 convergence_theorems: 0.99,
 product_measure_relation: 0.99,
 Radon_Nikodym_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(underlying_set_sigma_algebra_and_measure)
 -> determine(set_or_function_measurability)
 -> identify(null_sets_and_almost_everywhere_context)
 -> analyse(Lebesgue_integrability)
 -> identify(convergence_mode)
 -> select(valid_convergence_theorem)
 -> analyse(product_measure_if_present)
 -> apply(Fubini_or_Tonelli_if_hypotheses_hold)
 -> analyse(signed_measure_and_absolute_continuity)
 -> apply(Radon_Nikodym_if_valid)
 -> validate(measurability_integrability_and_sigma_finiteness)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed measure-theory semantic graph",
 primary_entity: MEASURE_THEORY,
 preserve: [
  underlying_set,
  sigma_algebra,
  measure,
  measurable_sets,
  measurable_functions,
  null_sets,
  convergence_mode,
  integrability
 ],
 infer_through: [
  outer_measures,
  Lebesgue_measure,
  Lebesgue_integrals,
  convergence_theorems,
  product_measures,
  signed_measures,
  absolute_continuity,
  Radon_Nikodym_derivatives
 ],
 avoid: [
  assigning_measure_to_nonmeasurable_set_without_context,
  confusing_almost_everywhere_with_everywhere,
  interchanging_limit_and_integral_without_hypotheses,
  applying_Fubini_without_integrability,
  confusing_outer_measure_with_measure,
  applying_Radon_Nikodym_without_absolute_continuity
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + MEASURE_THEORY
 + SIGMA_ALGEBRA
 + MEASURE
 + MEASURE_SPACE
 + MEASURABLE_FUNCTION
 + LEBESGUE_MEASURE
 + LEBESGUE_INTEGRAL
 + PRODUCT_MEASURE
 + ABSOLUTE_CONTINUITY
 + RADON_NIKODYM_DERIVATIVE
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> MEASURE_THEORY
 -> SPACE_SIGMA_MEASURE_IDENTIFICATION
 -> MEASURABILITY_ANALYSIS
 -> NULL_SET_ANALYSIS
 -> LEBESGUE_INTEGRATION_ANALYSIS
 -> CONVERGENCE_MODE_ANALYSIS
 -> CONVERGENCE_THEOREM_APPLICATION
 -> PRODUCT_MEASURE_ANALYSIS
 -> FUBINI_TONELLI_ANALYSIS
 -> SIGNED_MEASURE_ANALYSIS
 -> RADON_NIKODYM_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW