Lebesgue Integral Ontology

flow "LebesgueIntegralOntology" v1.0

# ============================================================
# LEBESGUE INTEGRAL ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
LEBESGUE_INTEGRAL := {
    type: MeasureTheoreticIntegral,
    domain: MeasureTheory,
    purpose: "Integrate measurable functions with respect to a measure by building from simple functions and supporting powerful limit and convergence theorems."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    LEBESGUE_INTEGRAL,
    MEASURE_SPACE,
    MEASURABLE_FUNCTION,
    SIMPLE_FUNCTION,
    INDICATOR_FUNCTION,
    NONNEGATIVE_FUNCTION,
    POSITIVE_PART,
    NEGATIVE_PART,
    INTEGRABLE_FUNCTION,
    L1_SPACE,
    ALMOST_EVERYWHERE,
    MONOTONE_CONVERGENCE,
    FATOU_LEMMA,
    DOMINATED_CONVERGENCE,
    TONELLI_THEOREM,
    FUBINI_THEOREM
}

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

LEBESGUE_INTEGRAL
    -> defined_relative_to(MEASURE_mu)

notation:

integral_X(
    f dmu
)

# ---------- INDICATOR FUNCTION ----------
for measurable
    A in Sigma

indicator_A(x)
    :=
1 if x in A
0 otherwise

integral_X(
    indicator_A dmu
)
=
mu(A)

INDICATOR_INTEGRAL
    -> recovers(MEASURE)

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

where
    a_k >= 0
    and
    A_k measurable

NONNEGATIVE_SIMPLE_FUNCTION
    -> finite_measurable_range

# ---------- SIMPLE FUNCTION INTEGRAL ----------
integral_X(
    phi dmu
)
    :=
sum_from_k_1_to_n(
    a_k
    *
    mu(A_k)
)

for disjoint representation

SIMPLE_FUNCTION_INTEGRAL
    -> finite_sum_of(VALUE_TIMES_MEASURE)

# ---------- NONNEGATIVE FUNCTION ----------
let
    f:X -> [0,infinity]

with
    f measurable

define

integral_X(
    f dmu
)
    :=
sup(
    integral_X(
        phi dmu
    )
)

over all
    measurable_simple_phi

such_that

0 <= phi <= f

NONNEGATIVE_LEBESGUE_INTEGRAL
    -> extended_real_value_in([0,infinity])

# ---------- SIMPLE APPROXIMATION ----------
for nonnegative measurable f:

exists sequence
    phi_n

such_that

0 <= phi_1 <= phi_2 <= ...

and

phi_n(x)
    -> f(x)

pointwise

SIMPLE_FUNCTIONS
    -> approximate(MEASURABLE_FUNCTION_FROM_BELOW)

# ---------- POSITIVE / NEGATIVE PART ----------
for real-valued measurable f:

f_positive
    :=
max(f,0)

f_negative
    :=
max(-f,0)

then

f
    =
f_positive
-
f_negative

and

|f|
    =
f_positive
+
f_negative

# ---------- GENERAL INTEGRAL ----------
if not both

integral(f_positive dmu)
and
integral(f_negative dmu)

are infinite

then

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

GENERAL_LEBESGUE_INTEGRAL
    -> defined_via(POSITIVE_AND_NEGATIVE_PARTS)

# ---------- INTEGRABILITY ----------
f
    -> LEBESGUE_INTEGRABLE

iff

integral_X(
    |f| dmu
)
    < infinity

equivalently

integral(f_positive dmu)
    < infinity

and

integral(f_negative dmu)
    < infinity

INTEGRABILITY
    -> ABSOLUTE_INTEGRABILITY

# ---------- L1 SPACE ----------
L1(mu)
    := {
        measurable f :
        integral(
            |f| dmu
        )
        < infinity
    }
    modulo(ALMOST_EVERYWHERE_EQUALITY)

norm:

||f||_1
    :=
integral(
    |f| dmu
)

L1
    -> BANACH_SPACE

# ---------- ALMOST EVERYWHERE EQUIVALENCE ----------
if

f = g
    almost_everywhere

then

integral(
    f dmu
)
=
integral(
    g dmu
)

when integrals are appropriately defined

LEBESGUE_INTEGRAL
    -> insensitive_to(NULL_SET_MODIFICATIONS)

# ---------- LINEARITY ----------
if
    f,g integrable

and
    alpha,beta scalars

then

integral(
    alpha*f
    +
    beta*g
    dmu
)
=
alpha*integral(f dmu)
+
beta*integral(g dmu)

LEBESGUE_INTEGRAL
    -> LINEAR_ON(L1)

# ---------- MONOTONICITY ----------
if

f <= g
    almost_everywhere

and integrals defined

then

integral(f dmu)
    <=
integral(g dmu)

MONOTONICITY
    -> order_preserving_property

# ---------- ABSOLUTE VALUE INEQUALITY ----------
if
    f integrable

then

|
integral(f dmu)
|
<=
integral(
    |f| dmu
)

INTEGRAL_TRIANGLE_INEQUALITY
    -> controls(INTEGRAL_MAGNITUDE)

# ---------- 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 LEMMA ----------
if
    f_n >= 0
    measurable

then

integral(
    liminf_n f_n
    dmu
)
<=
liminf_n(
    integral(
        f_n dmu
    )
)

FATOU_LEMMA
    -> lower_semicontinuity_of(NONNEGATIVE_INTEGRAL)

# ---------- DOMINATED CONVERGENCE ----------
if

f_n
    -> f
       almost_everywhere

and

|f_n|
    <= g
       almost_everywhere

with

g in L1(mu)

then

f in L1(mu)

and

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

DOMINATED_CONVERGENCE_THEOREM
    -> limit_integral_interchange_under(INTEGRABLE_DOMINATION)

# ---------- L1 CONVERGENCE CONSEQUENCE ----------
under dominated convergence hypotheses:

integral(
    |f_n-f|
    dmu
)
    -> 0

therefore

f_n
    -> f
       in(L1)

DOMINATION
    -> supports(L1_CONVERGENCE)

# ---------- BOUNDED CONVERGENCE ----------
on finite measure space:

if
    f_n -> f almost_everywhere

and
    |f_n| <= M

uniformly in n

then

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

BOUNDED_CONVERGENCE
    -> special_case_of(DOMINATED_CONVERGENCE)

# ---------- INTEGRAL OVER SET ----------
for measurable A:

integral_A(
    f dmu
)
    :=
integral_X(
    f
    *
    indicator_A
    dmu
)

SET_RESTRICTED_INTEGRAL
    -> localisation_of(INTEGRATION)

# ---------- ABSOLUTE CONTINUITY OF INTEGRAL ----------
if
    f in L1(mu)

then for every epsilon>0
there exists delta>0

such_that

mu(A)<delta

implies

integral_A(
    |f| dmu
)
<
epsilon

under standard finite-measure-control formulation

LEBESGUE_INTEGRAL
    -> absolutely_continuous_with_respect_to(MEASURE)

# ---------- RIESZ-FISCHER / COMPLETENESS ----------
if
    f_n
        -> Cauchy_in(L1)

then

exists
    f in L1

such_that

||f_n-f||_1
    -> 0

L1
    -> COMPLETE_NORMED_SPACE

# ---------- RIEMANN RELATION ----------
if
    f
        -> Riemann_integrable_on([a,b])

then

f
    -> Lebesgue_integrable

and

Riemann_integral
    =
Lebesgue_integral

LEBESGUE_INTEGRAL
    -> extends(RIEMANN_INTEGRAL)

# ---------- LEBESGUE CRITERION FOR RIEMANN INTEGRABILITY ----------
bounded function
    f:[a,b]->R

is Riemann integrable

iff

set_of_discontinuities
    -> Lebesgue_measure_zero

LEBESGUE_CRITERION
    -> connects(RIEMANN_AND_MEASURE_THEORETIC_INTEGRATION)

# ---------- PRODUCT SPACES ----------
measure spaces:

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

product measure:

mu tensor nu

function:

f:X x Y -> extended_real_numbers

PRODUCT_INTEGRATION
    -> governed_by(TONELLI_AND_FUBINI)

# ---------- TONELLI ----------
if
    f >= 0
    measurable

then

integral_(XxY)(
    f d(mu tensor nu)
)
=
integral_X(
    integral_Y(
        f(x,y) dnu(y)
    )
    dmu(x)
)

=
reverse_iterated_integral

possibly infinite

TONELLI_THEOREM
    -> permits(INTEGRATION_ORDER_FOR_NONNEGATIVE_FUNCTIONS)

# ---------- FUBINI ----------
if
    f in L1(mu tensor nu)

then

iterated_integrals
    -> finite_almost_everywhere_as_required

and

product_integral
    =
iterated_integrals

FUBINI_THEOREM
    -> permits(INTEGRATION_ORDER_FOR_INTEGRABLE_FUNCTIONS)

# ---------- CHANGE OF VARIABLES ----------
for suitable transformation
    T

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

CHANGE_OF_VARIABLES
    -> transforms(LEBESGUE_INTEGRAL_BY_JACOBIAN)

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

T:X->Y

pushforward measure:

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

then for suitable measurable g:

integral_Y(
    g(y)
    d(T_sharp mu)(y)
)
=
integral_X(
    g(T(x))
    dmu(x)
)

PUSHFORWARD
    -> measure_theoretic_change_of_variables

# ---------- EXPECTATION ----------
probability space:

(Omega,F,P)

random variable
    X

EXPECTATION:

E[X]
    :=
integral_Omega(
    X dP
)

EXPECTATION
    -> LEBESGUE_INTEGRAL_WITH_RESPECT_TO(PROBABILITY_MEASURE)

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

# ---------- RELATION GRAPH ----------
MEASURE_SPACE
    -> supports(MEASURABLE_FUNCTIONS)

MEASURABLE_FUNCTION
    -> approximated_by(SIMPLE_FUNCTIONS)

SIMPLE_FUNCTION
    -> has(DIRECT_MEASURE_BASED_INTEGRAL)

NONNEGATIVE_FUNCTION
    -> integrated_by(SUPREMUM_OF_SIMPLE_INTEGRALS)

SIGNED_FUNCTION
    -> decomposed_into(POSITIVE_AND_NEGATIVE_PARTS)

INTEGRABILITY
    -> equivalent_to(FINITE_ABSOLUTE_INTEGRAL)

NULL_SET
    -> ignored_by(ALMOST_EVERYWHERE_INTEGRATION)

CONVERGENCE_THEOREMS
    -> control(LIMIT_INTEGRAL_INTERCHANGE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Lebesgue integral" ~= "measure-theoretic integral defined from measurable simple functions",
    "Lebesgue integrable" ~= "measurable function with finite integral of its absolute value",
    "simple function integral" ~= "weighted sum of measures of measurable level sets",
    "monotone convergence theorem" ~= "interchange of increasing nonnegative limit and integral",
    "dominated convergence theorem" ~= "interchange of limit and integral under integrable domination",
    "integral over a measurable set" ~= "integral of the function multiplied by the set indicator"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    integrand_must_be_measurable_for_standard_Lebesgue_integral: true,
    integrability_requires_finite_absolute_integral: true,
    almost_everywhere_equal_integrable_functions_have_same_integral: true,
    dominated_convergence_requires_integrable_dominating_function: true,
    Tonelli_and_Fubini_have_distinct_hypotheses: true,
    positive_and_negative_parts_must_not_both_have_infinite_integral_for_signed_extended_integral: true,
    change_of_variables_requires_transformation_specific_hypotheses: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    simple_function_relation: 0.99,
    integrability_relation: 0.99,
    convergence_theorem_relation: 0.99,
    Fubini_Tonelli_relation: 0.99,
    probability_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(measure_space_and_integrand)
    -> verify(measurability)
    -> classify(nonnegative_signed_or_absolutely_integrable_case)
    -> construct(simple_function_approximation_if_needed)
    -> evaluate_or_bound(integral)
    -> identify(almost_everywhere_equivalences)
    -> select(monotone_Fatou_dominated_or_bounded_convergence_theorem_if_relevant)
    -> analyse(product_space_with_Tonelli_or_Fubini_if_present)
    -> apply(change_of_variables_or_pushforward_if_requested)
    -> validate(integrability_domination_and_measure_hypotheses)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Lebesgue-integral semantic graph",
    primary_entity: LEBESGUE_INTEGRAL,
    preserve: [
        measure_space,
        measure,
        integrand,
        measurability,
        positive_negative_parts,
        integrability,
        almost_everywhere_context,
        convergence_mode
    ],
    infer_through: [
        indicator_integrals,
        simple_functions,
        nonnegative_integrals,
        L1_spaces,
        monotone_convergence,
        Fatou_lemma,
        dominated_convergence,
        Tonelli_and_Fubini
    ],
    avoid: [
        integrating_nonmeasurable_function_without extended context,
        treating_conditional_improper_integrability_as L1_integrability,
        interchanging_limit_and_integral_without theorem hypotheses,
        applying_Fubini_when_only_Tonelli_conditions_hold,
        confusing_pointwise_equality_with almost_everywhere equality,
        ignoring_positive_negative_infinite_part ambiguity
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + LEBESGUE_INTEGRAL
    + MEASURE_SPACE
    + MEASURABLE_FUNCTION
    + SIMPLE_FUNCTION
    + INTEGRABLE_FUNCTION
    + L1_SPACE
    + MONOTONE_CONVERGENCE
    + FATOU_LEMMA
    + DOMINATED_CONVERGENCE
    + TONELLI_THEOREM
    + FUBINI_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> LEBESGUE_INTEGRAL
    -> MEASURE_SPACE_INTEGRAND_IDENTIFICATION
    -> MEASURABILITY_ANALYSIS
    -> FUNCTION_TYPE_CLASSIFICATION
    -> SIMPLE_APPROXIMATION_ANALYSIS
    -> INTEGRAL_EVALUATION
    -> ALMOST_EVERYWHERE_ANALYSIS
    -> CONVERGENCE_THEOREM_ANALYSIS
    -> PRODUCT_SPACE_ANALYSIS
    -> CHANGE_VARIABLE_PUSHFORWARD_ANALYSIS
    -> HYPOTHESIS_VALIDATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW