Lebesgue Measure Ontology

flow "LebesgueMeasureOntology" v1.0

# ============================================================
# LEBESGUE MEASURE ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
LEBESGUE_MEASURE := {
    type: TranslationInvariantMeasure,
    domain: MeasureTheory,
    purpose: "Assign length, area, volume and higher-dimensional Euclidean size to Lebesgue measurable subsets of Euclidean space."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    LEBESGUE_MEASURE,
    LEBESGUE_MEASURABLE_SET,
    LEBESGUE_SIGMA_ALGEBRA,
    BOREL_SIGMA_ALGEBRA,
    OUTER_MEASURE,
    INTERVAL,
    RECTANGLE,
    NULL_SET,
    COMPLETION,
    TRANSLATION_INVARIANCE,
    COUNTABLE_ADDITIVITY,
    REGULARITY,
    PRODUCT_MEASURE,
    ALMOST_EVERYWHERE
}

# ---------- REAL LINE ----------
on
    R

Lebesgue measure
    := lambda

for interval:

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

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

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

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

ENDPOINTS
    -> have(ZERO_LEBESGUE_MEASURE)

# ---------- SINGLETON ----------
for
    x in R

lambda({x})
    = 0

SINGLETON
    -> LEBESGUE_NULL_SET

COUNTABLE_SET
    -> measure_zero
       in(R^n)

# ---------- EUCLIDEAN SPACE ----------
on
    R^n

lambda_n
    -> n_DIMENSIONAL_LEBESGUE_MEASURE

for rectangle

R
    =
product_from_i_1_to_n(
    [a_i,b_i]
)

lambda_n(R)
    =
product_from_i_1_to_n(
    b_i-a_i
)

LEBESGUE_MEASURE
    -> generalises(LENGTH_AREA_VOLUME)

# ---------- OUTER MEASURE CONSTRUCTION ----------
for
    E subset_of R^n

Lebesgue_outer_measure:

lambda_star(E)
    :=
inf(
    sum_k(
        volume(Q_k)
    )
)

over countable coverings
    E subset_of union_k(Q_k)

where
    Q_k
        -> boxes_or_rectangles

OUTER_MEASURE
    -> defined_on(ALL_SUBSETS)

# ---------- CARATHEODORY MEASURABILITY ----------
E
    -> LEBESGUE_MEASURABLE

if

for every
    A subset_of R^n

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

CARATHEODORY_CRITERION
    -> defines(LEBESGUE_MEASURABLE_SETS)

# ---------- LEBESGUE SIGMA ALGEBRA ----------
L
    :=
{
    E subset_of R^n :
    E Lebesgue_measurable
    }

L
    -> SIGMA_ALGEBRA

LEBESGUE_MEASURE
    :=
lambda_star restricted_to L

# ---------- BOREL RELATION ----------
BOREL_SIGMA_ALGEBRA
    subset_of
LEBESGUE_SIGMA_ALGEBRA

every
    BOREL_SET
    -> LEBESGUE_MEASURABLE

but

LEBESGUE_SIGMA_ALGEBRA
    -> generally_strictly_larger_than(BOREL_SIGMA_ALGEBRA)

# ---------- COMPLETION ----------
LEBESGUE_SIGMA_ALGEBRA
    -> completion_of(BOREL_SIGMA_ALGEBRA)
       with_respect_to(LEBESGUE_MEASURE)

if
    N
        -> BOREL_OR_LEBESGUE_NULL_SET

then every
    A subset_of N

-> LEBESGUE_MEASURABLE
-> lambda(A)=0

LEBESGUE_MEASURE
    -> COMPLETE

# ---------- COUNTABLE ADDITIVITY ----------
if
    E_1,E_2,...
        -> pairwise_disjoint_Lebesgue_measurable_sets

then

lambda(
    union_n(E_n)
)
=
sum_n(
    lambda(E_n)
)

LEBESGUE_MEASURE
    -> COUNTABLY_ADDITIVE

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

and both measurable

then

lambda(A)
    <=
lambda(B)

MONOTONICITY
    -> fundamental_measure_property

# ---------- TRANSLATION INVARIANCE ----------
for
    E measurable

and
    a in R^n

E+a
    :=
{
    x+a :
    x in E
    }

lambda(E+a)
    =
lambda(E)

LEBESGUE_MEASURE
    -> TRANSLATION_INVARIANT

# ---------- ROTATION INVARIANCE ----------
for orthogonal transformation
    Q

lambda(QE)
    =
lambda(E)

EUCLIDEAN_ISOMETRY
    -> preserves(LEBESGUE_MEASURE)

# ---------- SCALING ----------
for scalar
    c

lambda_n(cE)
    =
|c|^n
*
lambda_n(E)

SCALING
    -> dimension_dependent

more generally for invertible linear map
    A:

lambda_n(AE)
    =
|det(A)|
*
lambda_n(E)

# ---------- CHANGE OF VARIABLES ----------
for suitable differentiable bijection
    T

integral_{T(E)}(
    f(y)
    dy
)
=
integral_E(
    f(T(x))
    *
    |det(DT(x))|
    dx
)

JACOBIAN
    -> local_measure_scaling_factor

LEBESGUE_MEASURE
    -> supports(CHANGE_OF_VARIABLES)

# ---------- NULL SET ----------
E
    -> NULL_SET

if

lambda(E)
    = 0

NULL_SET
    -> negligible_for(LEBESGUE_INTEGRATION)

COUNTABLE_UNION_OF_NULL_SETS
    -> NULL_SET

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

if

lambda(
    {x : P(x) fails}
)
=
0

then

P
    -> holds(LEBESGUE_ALMOST_EVERYWHERE)

ALMOST_EVERYWHERE
    -> ignores(NULL_SET_EXCEPTIONS)

# ---------- DENSE NULL SET ----------
RATIONAL_NUMBERS
    -> COUNTABLE
    -> LEBESGUE_MEASURE_ZERO
    -> DENSE_IN(R)

DENSITY
    -/> implies(POSITIVE_MEASURE)

TOPOLOGICAL_SIZE
    -/> same_as(MEASURE_THEORETIC_SIZE)

# ---------- OPEN SET REGULARITY ----------
for measurable E:

lambda(E)
    =
inf(
    lambda(O)
    :
    E subset_of O,
    O open
)

OUTER_REGULARITY
    -> approximation_from_outside_by(OPEN_SETS)

# ---------- INNER REGULARITY ----------
for measurable E of finite measure:

lambda(E)
    =
sup(
    lambda(K)
    :
    K subset_of E,
    K compact
)

INNER_REGULARITY
    -> approximation_from_inside_by(COMPACT_SETS)

LEBESGUE_MEASURE
    -> REGULAR_MEASURE

# ---------- CONTINUITY FROM BELOW ----------
if

E_1 subset_of E_2 subset_of ...

then

lambda(
    union_n E_n
)
=
lim_n(
    lambda(E_n)
)

# ---------- CONTINUITY FROM ABOVE ----------
if

E_1 superset_of E_2 superset_of ...

and

lambda(E_1)
    < infinity

then

lambda(
    intersection_n E_n
)
=
lim_n(
    lambda(E_n)
)

# ---------- PRODUCT MEASURE ----------
Lebesgue measure on
    R^(m+n)

-> agrees_with(
    lambda_m tensor lambda_n
)

under standard identification

PRODUCT_LEBESGUE_MEASURE
    -> supports(FUBINI_AND_TONELLI)

# ---------- INTEGRATION ----------
for measurable
    f:R^n -> extended_real_numbers

integral(
    f(x)
    d lambda_n(x)
)

usually written

integral(
    f(x)
    dx
)

dx
    -> notation_for(LEBESGUE_MEASURE_INTEGRATION)

# ---------- INDICATOR FUNCTION ----------
for measurable E:

integral(
    indicator_E
    d lambda
)
=
lambda(E)

MEASURE
    -> integral_of(INDICATOR_FUNCTION)

# ---------- PROBABILITY DENSITY RELATION ----------
if
    probability measure P
        -> absolutely_continuous_with_respect_to(lambda)

then

exists density
    p(x)

such_that

P(E)
    =
integral_E(
    p(x)
    dx
)

LEBESGUE_MEASURE
    -> reference_measure_for(CONTINUOUS_PROBABILITY_DISTRIBUTIONS)

# ---------- NONMEASURABLE SETS ----------
there exist
    subsets_of(R)

that are not
    Lebesgue_measurable

VITALI_TYPE_CONSTRUCTION
    -> example_of(NONMEASURABLE_SET)

LEBESGUE_MEASURE
    -/> defined_as_countably_additive_translation_invariant_length_on_ALL_SUBSETS)

MEASURABILITY_RESTRICTION
    -> mathematically_necessary

# ---------- UNIQUENESS CHARACTERISATION ----------
on R^n:

Lebesgue measure
    -> translation_invariant
    -> locally_finite
    -> normalised_by(unit_cube_measure=1)

and is unique under
    appropriate_regular_Borel_measure_conditions

LEBESGUE_MEASURE
    -> canonical_Euclidean_volume_measure

# ---------- APPLICATIONS ----------
LEBESGUE_MEASURE
    -> applied_to(LEBESGUE_INTEGRATION)
    -> applied_to(REAL_ANALYSIS)
    -> applied_to(PROBABILITY)
    -> applied_to(FOURIER_ANALYSIS)
    -> applied_to(PDEs)
    -> applied_to(GEOMETRIC_MEASURE_THEORY)
    -> applied_to(FUNCTIONAL_ANALYSIS)

# ---------- RELATION GRAPH ----------
OUTER_MEASURE
    -> constructs(LEBESGUE_MEASURABLE_SETS)

LEBESGUE_SIGMA_ALGEBRA
    -> domain_of(LEBESGUE_MEASURE)

LEBESGUE_MEASURE
    -> extends(EUCLIDEAN_LENGTH_AREA_VOLUME)

LEBESGUE_MEASURE
    -> defines(NULL_SETS)

NULL_SET
    -> defines(ALMOST_EVERYWHERE)

LEBESGUE_MEASURE
    -> supports(LEBESGUE_INTEGRAL)

TRANSLATION
    -> preserves(LEBESGUE_MEASURE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Lebesgue measure" ~= "standard translation-invariant Euclidean measure extending length, area and volume",
    "Lebesgue measurable set" ~= "set satisfying the Lebesgue-Caratheodory measurability condition",
    "Lebesgue null set" ~= "Lebesgue measurable set of measure zero",
    "Lebesgue sigma algebra" ~= "completion of the Borel sigma algebra under Lebesgue measure",
    "Lebesgue outer measure" ~= "outer measure obtained from countable coverings by intervals or boxes",
    "almost everywhere with respect to Lebesgue measure" ~= "outside a Lebesgue null set"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    Lebesgue_measure_is_defined_on_Lebesgue_measurable_sets_not_all_subsets: true,
    Borel_sets_are_Lebesgue_measurable_but_not_every_Lebesgue_measurable_set_is_Borel: true,
    Lebesgue_measure_is_complete: true,
    translation_invariance_must_be_preserved: true,
    scaling_in_Rn_uses_absolute_determinant_or_dimension_power: true,
    dense_sets_can_have_measure_zero: true,
    topological_size_and_measure_theoretic_size_are_distinct: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    outer_measure_relation: 0.99,
    Borel_completion_relation: 0.99,
    invariance_relation: 0.99,
    null_set_relation: 0.99,
    integration_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(Euclidean_space_and_dimension)
    -> identify(candidate_set)
    -> determine(Lebesgue_measurability)
    -> compute_or_bound(outer_measure_if_needed)
    -> apply(interval_rectangle_translation_or_scaling_properties)
    -> identify(null_set_and_almost_everywhere_structure)
    -> use(regularity_for_open_or_compact_approximation_if_relevant)
    -> connect_to(Lebesgue_integral_or_probability_density_if_requested)
    -> analyse(product_measure_or_change_of_variables_if_present)
    -> validate(measurability_dimension_and_finiteness_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Lebesgue-measure semantic graph",
    primary_entity: LEBESGUE_MEASURE,
    preserve: [
        Euclidean_space,
        dimension,
        measurable_set,
        outer_measure,
        null_set_status,
        transformations,
        finiteness,
        sigma_algebra
    ],
    infer_through: [
        interval_measure,
        rectangle_measure,
        Caratheodory_measurability,
        Borel_completion,
        translation_invariance,
        scaling,
        regularity,
        product_measure
    ],
    avoid: [
        assigning_Lebesgue_measure_to_nonmeasurable_set_as ordinary measure,
        confusing_Borel_and_Lebesgue_sigma_algebras,
        assuming_dense_set_has positive measure,
        omitting_absolute_determinant_in linear change of variables,
        confusing_outer_measure_with measure,
        assuming_all_null_sets_are topologically small
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + LEBESGUE_MEASURE
    + LEBESGUE_MEASURABLE_SET
    + LEBESGUE_SIGMA_ALGEBRA
    + OUTER_MEASURE
    + NULL_SET
    + COMPLETION
    + TRANSLATION_INVARIANCE
    + PRODUCT_MEASURE
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> LEBESGUE_MEASURE
    -> SPACE_DIMENSION_IDENTIFICATION
    -> SET_MEASURABILITY_ANALYSIS
    -> OUTER_MEASURE_ANALYSIS
    -> INVARIANCE_SCALING_ANALYSIS
    -> NULL_SET_ANALYSIS
    -> REGULARITY_ANALYSIS
    -> INTEGRATION_PROBABILITY_RELATION
    -> PRODUCT_CHANGE_VARIABLE_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW