Sigma Algebra Ontology

flow "SigmaAlgebraOntology" v1.0

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

SIGMA_ALGEBRA := {
 type: MeasurableSetStructure,
 domain: MeasureTheory,
 purpose: "Define a collection of subsets closed under complements and countable unions, providing the measurable domain on which measures and measurable functions are defined."
}

CONCEPTS := {
 SIGMA_ALGEBRA,
 UNDERLYING_SET,
 POWER_SET,
 MEASURABLE_SET,
 MEASURABLE_SPACE,
 GENERATED_SIGMA_ALGEBRA,
 BOREL_SIGMA_ALGEBRA,
 TRIVIAL_SIGMA_ALGEBRA,
 POWER_SET_SIGMA_ALGEBRA,
 COMPLETION,
 PRODUCT_SIGMA_ALGEBRA,
 MEASURABLE_FUNCTION,
 MEASURE
}

# ---------- DEFINITION ----------
let
 X -> UNDERLYING_SET

Sigma
 subset_of POWER_SET(X)

Sigma
 -> SIGMA_ALGEBRA_ON(X)

iff

X in Sigma

AND

A in Sigma
 -> X\A in Sigma

AND

A_1,A_2,... in Sigma
 -> union_from_n_1_to_infinity(A_n) in Sigma

# ---------- EMPTY SET ----------
X in Sigma

and

X\X
 = empty_set

therefore

empty_set
 in Sigma

SIGMA_ALGEBRA
 -> contains(EMPTY_SET)

# ---------- COUNTABLE INTERSECTIONS ----------
A_n in Sigma

-> X\A_n in Sigma

-> union_n(X\A_n) in Sigma

by De Morgan:

X\union_n(X\A_n)
 =
intersection_n(A_n)

therefore

intersection_n(A_n)
 in Sigma

SIGMA_ALGEBRA
 -> closed_under(COUNTABLE_INTERSECTIONS)

# ---------- FINITE OPERATIONS ----------
COUNTABLE_UNION_CLOSURE
 -> FINITE_UNION_CLOSURE

COUNTABLE_INTERSECTION_CLOSURE
 -> FINITE_INTERSECTION_CLOSURE

A,B in Sigma
 -> A union B in Sigma
 -> A intersection B in Sigma
 -> A\B in Sigma

SIGMA_ALGEBRA
 -> closed_under(STANDARD_SET_OPERATIONS)

# ---------- MEASURABLE SET ----------
A
 -> MEASURABLE_SET

relative_to Sigma

iff

A in Sigma

MEASURABILITY
 -> structure_dependent

same subset
 -> may_be_measurable_under_one_sigma_algebra
 -> may_not_be_measurable_under_another

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

SIGMA_ALGEBRA
 -> supplies(MEASURABLE_STRUCTURE)

MEASURABLE_SPACE
 -> prerequisite_for(MEASURE)

# ---------- TRIVIAL SIGMA ALGEBRA ----------
TRIVIAL_SIGMA_ALGEBRA
 := {
 empty_set,
 X
 }

TRIVIAL_SIGMA_ALGEBRA
 -> smallest_sigma_algebra_on_nonempty_X

# ---------- POWER SET SIGMA ALGEBRA ----------
POWER_SET(X)
 -> SIGMA_ALGEBRA

POWER_SET_SIGMA_ALGEBRA
 -> largest_sigma_algebra_on(X)

if X countable:
 POWER_SET
 -> commonly_supports(DISCRETE_MEASURES)

# ---------- INTERSECTION PROPERTY ----------
let
 Sigma_i
 -> family_of_sigma_algebras_on(X)

then

intersection_i(Sigma_i)
 -> SIGMA_ALGEBRA

INTERSECTION_PROPERTY
 -> enables(GENERATED_SIGMA_ALGEBRA)

# ---------- GENERATED SIGMA ALGEBRA ----------
for collection
 C subset_of POWER_SET(X)

sigma(C)
 :=
intersection_of(
 all_sigma_algebras_containing(C)
 )

sigma(C)
 -> SMALLEST_SIGMA_ALGEBRA_CONTAINING(C)

C
 -> GENERATOR

# ---------- FINITE GENERATOR ----------
if
 C
 -> finite_partition_or_finite_collection

then

sigma(C)
 -> generated_by(
 complements,
 unions,
 intersections
 )

FINITE_GENERATION
 -> may_produce(FINITE_SIGMA_ALGEBRA)

# ---------- ATOMS ----------
ATOM
 -> minimal_nonempty_measurable_component
    in appropriate finite_or_atomic context

FINITE_SIGMA_ALGEBRA
 -> unions_of(ATOMS)

ATOMIC_STRUCTURE
 -> simplifies(MEASURE_ASSIGNMENT)

# ---------- BOREL SIGMA ALGEBRA ----------
for topological space
 (X,Tau)

BOREL_SIGMA_ALGEBRA
 := sigma(Tau)

BOREL_SIGMA_ALGEBRA
 -> generated_by(OPEN_SETS)

equivalently
 -> generated_by(CLOSED_SETS)

BOREL_SET
 -> element_of(BOREL_SIGMA_ALGEBRA)

# ---------- REAL LINE BOREL STRUCTURE ----------
on R:

B(R)
 -> generated_by(OPEN_INTERVALS)

also generated by:
 CLOSED_INTERVALS
 HALF_OPEN_INTERVALS
 OPEN_RAYS
 CLOSED_RAYS
 RATIONAL_ENDPOINT_INTERVALS

COUNTABLE_BASE
 -> useful_for(BOREL_GENERATION)

# ---------- LEBESGUE SIGMA ALGEBRA ----------
LEBESGUE_SIGMA_ALGEBRA
 -> completion_of(BOREL_SIGMA_ALGEBRA_WITH_RESPECT_TO_LEBESGUE_MEASURE)

BOREL_SIGMA_ALGEBRA
 subset_of
LEBESGUE_SIGMA_ALGEBRA

LEBESGUE_MEASURABLE_SET
 -> may_not_be(BOREL_SET)

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

COMPLETED_SIGMA_ALGEBRA
 -> includes(
 Sigma
 plus
 all_subsets_of_mu_null_sets
 )

COMPLETION
 -> makes(MEASURE_SPACE_COMPLETE)

# ---------- SUB-SIGMA ALGEBRA ----------
if
 G subset_of Sigma

and
G itself sigma_algebra

then

G
 -> SUB_SIGMA_ALGEBRA

SUB_SIGMA_ALGEBRA
 -> represents(COARSER_INFORMATION)

in probability:

G
 -> INFORMATION_SET

# ---------- FILTRATION ----------
FILTRATION
 := {
 F_t
 }

such_that

s <= t
 -> F_s subset_of F_t

each
 F_t
 -> SIGMA_ALGEBRA

FILTRATION
 -> increasing_information_structure

PROBABILITY
 -> uses(FILTRATIONS_FOR_STOCHASTIC_PROCESSES)

# ---------- PRODUCT SIGMA ALGEBRA ----------
measurable spaces:

(X,Sigma)
(Y,Tau)

PRODUCT_SIGMA_ALGEBRA
 :=
Sigma tensor Tau

:=
sigma(
 {
  A x B :
  A in Sigma,
  B in Tau
 }
)

PRODUCT_SIGMA_ALGEBRA
 -> measurable_structure_on(X x Y)

# ---------- CYLINDER SETS ----------
in product spaces:

CYLINDER_SET
 -> constrains(FINITE_COORDINATES)

CYLINDER_SETS
 -> often_generate(PRODUCT_SIGMA_ALGEBRA)

INFINITE_PRODUCT
 -> requires(CAREFUL_SIGMA_ALGEBRA_CONSTRUCTION)

# ---------- TRACE SIGMA ALGEBRA ----------
for
 E subset_of X

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

TRACE_SIGMA_ALGEBRA
 -> measurable_structure_inherited_by(SUBSET)

# ---------- PULLBACK SIGMA ALGEBRA ----------
given
 f:X -> Y

and sigma algebra
 Tau on Y

f_inverse(Tau)
 :=
{
 f_inverse(B) :
 B in Tau
 }

PULLBACK_SIGMA_ALGEBRA
 -> sigma_algebra_on(X)

MEASURABILITY
 -> expressed_via(PREIMAGES)

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

f
 -> MEASURABLE

iff

for_all(B in Tau):

f_inverse(B)
 in Sigma

SIGMA_ALGEBRA
 -> defines(FUNCTION_MEASURABILITY)

# ---------- INITIAL SIGMA ALGEBRA ----------
given maps
 f_i:X -> Y_i

INITIAL_SIGMA_ALGEBRA
 -> smallest_sigma_algebra_on(X)
    making_all(f_i)_measurable

GENERATED_BY_PREIMAGES
 -> information_required_for(MAPS_TO_BE_MEASURABLE)

# ---------- RANDOM VARIABLE ----------
on probability space
 (Omega,F,P)

random variable
 X:
 Omega -> R

X
 -> measurable_from(
 F,
 B(R)
 )

RANDOM_VARIABLE
 -> measurable_function

SIGMA(X)
 -> sigma_algebra_generated_by(RANDOM_VARIABLE)

# ---------- SIGMA ALGEBRA GENERATED BY RANDOM VARIABLE ----------
sigma(X)
 :=
{
 X_inverse(B) :
 B in B(R)
 }

sigma(X)
 -> INFORMATION_CONTAINED_IN(X)

MULTIPLE_RANDOM_VARIABLES
 -> generate(JOINT_INFORMATION_SIGMA_ALGEBRA)

# ---------- MEASURE DOMAIN ----------
measure

mu:
 Sigma -> [0,infinity]

SIGMA_ALGEBRA
 -> DOMAIN_OF_MEASURE

MEASURE
 -/> generally_defined_on(ALL_SUBSETS)

NONMEASURABLE_SETS
 -> outside(DOMAIN_OF_MEASURE)

# ---------- COUNTABILITY ROLE ----------
FINITE_ADDITIVITY
 -> insufficient_for(STANDARD_MEASURE_THEORY)

COUNTABLE_SET_OPERATIONS
 -> essential_for(
 LIMITS,
 CONVERGENCE,
 COUNTABLE_ADDITIVITY
 )

SIGMA_PREFIX
 -> indicates(COUNTABLE_CLOSURE)

# ---------- MONOTONE LIMITS OF SETS ----------
if
 A_n increasing
 and
 A_n in Sigma

then

union_n A_n
 in Sigma

if
 A_n decreasing

then

intersection_n A_n
 in Sigma

SIGMA_ALGEBRA
 -> stable_under(MONOTONE_SET_LIMITS)

# ---------- LIMSUP / LIMINF OF SETS ----------
limsup A_n
 :=
intersection_n(
 union_{k>=n} A_k
)

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

if
 A_n in Sigma

then

limsup A_n
 and
liminf A_n
 -> MEASURABLE

SIGMA_ALGEBRA
 -> supports(SET_LIMIT_OPERATIONS)

# ---------- APPLICATIONS ----------
SIGMA_ALGEBRA
 -> applied_to(MEASURE_THEORY)
 -> applied_to(PROBABILITY)
 -> applied_to(LEBESGUE_INTEGRATION)
 -> applied_to(STOCHASTIC_PROCESSES)
 -> applied_to(ERGODIC_THEORY)
 -> applied_to(FUNCTIONAL_ANALYSIS)
 -> applied_to(INFORMATION_STRUCTURES)

# ---------- RELATION GRAPH ----------
UNDERLYING_SET
 -> has(POWER_SET)

SIGMA_ALGEBRA
 -> subset_of(POWER_SET)

SIGMA_ALGEBRA
 -> defines(MEASURABLE_SETS)

SIGMA_ALGEBRA
 -> forms(MEASURABLE_SPACE)

MEASURE
 -> defined_on(SIGMA_ALGEBRA)

TOPOLOGY
 -> generates(BOREL_SIGMA_ALGEBRA)

NULL_SETS
 -> added_by(COMPLETION)

PRODUCT_SIGMA_ALGEBRA
 -> structures(PRODUCT_SPACE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "sigma algebra" ~= "collection of subsets closed under complements and countable unions",
 "measurable set" ~= "set belonging to the selected sigma algebra",
 "measurable space" ~= "underlying set equipped with a sigma algebra",
 "generated sigma algebra" ~= "smallest sigma algebra containing a specified family of sets",
 "Borel sigma algebra" ~= "sigma algebra generated by open sets of a topological space",
 "product sigma algebra" ~= "sigma algebra generated by measurable rectangles"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 sigma_algebra_is_collection_of_subsets_not_collection_of_points: true,
 closure_is_countable_not_arbitrary_uncountable: true,
 measurability_depends_on_selected_sigma_algebra: true,
 generated_sigma_algebra_is_smallest_sigma_algebra_containing_generators: true,
 Borel_and_Lebesgue_sigma_algebras_are_distinct: true,
 product_sigma_algebra_is_generated_by_measurable_rectangles: true,
 measure_is_defined_on_sigma_algebra_not_sigma_algebra_on_measure: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 closure_properties: 0.99,
 generated_structure: 0.99,
 Borel_relation: 0.99,
 measurable_function_relation: 0.99,
 product_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(underlying_set)
 -> identify(candidate_collection_or_generators)
 -> verify(empty_full_set_and_complement_closure)
 -> verify(countable_union_closure)
 -> infer(countable_intersection_and_difference_closure)
 -> construct(generated_sigma_algebra_if_requested)
 -> classify(trivial_power_set_Borel_completed_or_product_structure)
 -> analyse(measurable_sets_and_functions)
 -> connect_to(measure_or_probability_context_if_relevant)
 -> validate(countability_and_generator_relations)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed sigma-algebra semantic graph",
 primary_entity: SIGMA_ALGEBRA,
 preserve: [
  underlying_set,
  sigma_algebra,
  generators,
  measurable_sets,
  topology,
  completion_measure,
  product_factors
 ],
 infer_through: [
  complements,
  countable_unions,
  countable_intersections,
  generated_sigma_algebras,
  Borel_sigma_algebras,
  completions,
  product_sigma_algebras,
  measurable_functions
 ],
 avoid: [
  confusing_sigma_algebra_with ordinary algebraic_structure,
  assuming_every_subset_is measurable,
  replacing_countable_closure_with arbitrary_uncountable closure,
  confusing_Borel_and_Lebesgue measurability,
  defining_measure_before specifying measurable_domain,
  treating_product_sigma_algebra_as full_power_set_without justification
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + SIGMA_ALGEBRA
 + UNDERLYING_SET
 + MEASURABLE_SET
 + MEASURABLE_SPACE
 + GENERATED_SIGMA_ALGEBRA
 + BOREL_SIGMA_ALGEBRA
 + COMPLETION
 + PRODUCT_SIGMA_ALGEBRA
 + MEASURABLE_FUNCTION
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> SIGMA_ALGEBRA
 -> UNDERLYING_SET_IDENTIFICATION
 -> COLLECTION_GENERATOR_ANALYSIS
 -> AXIOM_VALIDATION
 -> CLOSURE_INFERENCE
 -> GENERATED_STRUCTURE_ANALYSIS
 -> SIGMA_TYPE_CLASSIFICATION
 -> MEASURABILITY_ANALYSIS
 -> MEASURE_PROBABILITY_RELATION
 -> COUNTABILITY_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW