{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.lebesgue-stieltjes-integration","code":"thing-q3799278","url":"https://ver.cy/models/thing/q3799278/","name":"Lebesgue–Stieltjes integration","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QLT"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"Lebesgue-Stieltjes integration is Lebesgue integration with respect to a measure induced by a monotone function, with extensions to signed or complex measures induced by functions of bounded variation.","purpose":"Enable an agent to recognise a Lebesgue-Stieltjes integral, record its generating function and conventions, and determine whether its evaluation or transformation is justified.","scope":{"in":["Construction of the measure associated with a generating function and a stated continuity convention","Measurability, integrability and the status of a proposed integral","Atomic, absolutely continuous and singular continuous contributions","Endpoint conventions, restrictions of the integration domain and unbounded domains","Conditions supporting convergence, integration by parts and comparison with Riemann-Stieltjes integration","Probability expectations expressed as integration against a distribution function"],"out":["General Lebesgue integration independent of a Stieltjes generating function","Riemann-Stieltjes integration as a separate theory","Itô, Stratonovich and other stochastic integration theories","Distributional derivatives and generalized functions beyond their relationship to the associated measure","Vector-valued integration and operator-valued measures","Multivariate Stieltjes constructions"],"boundaries":[]},"distinguishingFeatures":["The integrator determines a countably additive measure, and the integral is defined through Lebesgue integration against that measure.","For a finite-valued, nondecreasing, right-continuous generator F, the convention μ_F((a,b]) = F(b) − F(a) fixes interval masses.","A jump at x contributes an atom of mass F(x) − F(x−), so an integrand's value at that point can affect the integral.","Replacing dF by F'(x) dx is sufficient only when the relevant measure is absolutely continuous; jumps and singular continuous mass require additional treatment.","Existence does not depend on convergence of Riemann-Stieltjes partition sums, although the two integrals agree under suitable hypotheses."],"structure":{"bundles":[{"id":"generator-to-measure","name":"Generator to measure","description":"Record how the stated integrator determines the measure used by the integral.","layers":[{"id":"generator-admissibility","name":"Generator admissibility","description":"Establish the function class and locality of the construction.","findings":[{"id":"monotonicity-and-variation","name":"Monotonicity and variation","description":"Record whether F generates a positive measure or requires a qualified signed or complex extension.","questions":[{"text":"Is F finite-valued and nondecreasing, or is the construction based on bounded variation?","id":"monotonicity-and-variation-q01","kind":"definition"},{"text":"Are the required variation and finiteness assumptions local to the integration interval or valid on the entire domain?","id":"monotonicity-and-variation-q02","kind":"boundary"}]}]},{"id":"measure-convention","name":"Measure convention","description":"Specify the continuity representative, interval increments and measurable space.","findings":[{"id":"interval-increments","name":"Interval increments","description":"Make the correspondence between increments of F and interval masses explicit.","questions":[{"text":"Does the construction use right-continuous F with μ_F((a,b]) = F(b) − F(a), or another stated convention?","id":"interval-increments-q01","kind":"definition"},{"text":"Which reference fixes this convention, and is integration performed on the Borel σ-algebra or its completion?","id":"interval-increments-q02","kind":"provenance"}]}]}]},{"id":"integrand-and-existence","name":"Integrand and existence","description":"Determine whether the proposed integral is meaningful and what kind of value it can have.","layers":[{"id":"measurability-and-representatives","name":"Measurability and representatives","description":"Check admissible integrands and the null sets governing equivalence.","findings":[{"id":"measure-relative-integrand","name":"Measure-relative integrand","description":"Assess measurability and equality almost everywhere with respect to the associated measure.","questions":[{"text":"Is f measurable for the chosen measurable space and measure completion?","id":"measure-relative-integrand-q01","kind":"definition"},{"text":"Would changing f on a Lebesgue-null set alter its values on a set with positive μ_F mass?","id":"measure-relative-integrand-q02","kind":"boundary"}]}]},{"id":"integrability-status","name":"Integrability status","description":"Record the tests distinguishing finite, extended and undefined integrals.","findings":[{"id":"positive-negative-and-absolute-parts","name":"Positive, negative and absolute parts","description":"Use the appropriate positive-part or total-variation tests for the selected measure framework.","questions":[{"text":"For a positive μ_F and real f, what are ∫f⁺ dμ_F and ∫f⁻ dμ_F, and do both diverge?","id":"positive-negative-and-absolute-parts-q01","kind":"measurement"},{"text":"For a finite signed or complex measure, is ∫|f| d|μ_F| finite?","id":"positive-negative-and-absolute-parts-q02","kind":"measurement"}]}]}]},{"id":"mass-components-and-boundaries","name":"Mass components and boundaries","description":"Locate the mass that contributes to evaluation and specify which boundary points are included.","layers":[{"id":"atomic-and-endpoint-mass","name":"Atomic and endpoint mass","description":"Record jump contributions and the set denoted by an interval integral.","findings":[{"id":"jump-contributions","name":"Jump contributions","description":"Under the right-continuous convention, an included atom x contributes f(x)[F(x) − F(x−)].","questions":[{"text":"Which included points have nonzero jump mass, and what values of f are assigned there?","id":"jump-contributions-q01","kind":"measurement"},{"text":"Does the notation ∫ from a to b denote integration over (a,b], [a,b], [a,b) or another explicitly defined set?","id":"jump-contributions-q02","kind":"boundary"}]}]},{"id":"continuous-mass-components","name":"Continuous mass components","description":"Separate density-based mass from continuous mass singular to Lebesgue measure.","findings":[{"id":"density-and-singular-remainder","name":"Density and singular remainder","description":"Record the absolutely continuous contribution and any singular continuous remainder.","questions":[{"text":"What is the density of the absolutely continuous component with respect to Lebesgue measure?","id":"density-and-singular-remainder-q01","kind":"measurement"},{"text":"What evidence establishes that a singular continuous component is absent or must be retained?","id":"density-and-singular-remainder-q02","kind":"boundary"}]}]}]},{"id":"valid-transformations","name":"Valid transformations","description":"Track hypotheses permitting limits and algebraic transformations of the integral.","layers":[{"id":"limits-under-integration","name":"Limits under integration","description":"Justify convergence steps against a specified associated measure.","findings":[{"id":"convergence-hypotheses","name":"Convergence hypotheses","description":"Attach the relevant theorem and verified assumptions to a proposed interchange.","questions":[{"text":"For a fixed positive μ_F, do the integrands satisfy monotone convergence or dominated convergence hypotheses?","id":"convergence-hypotheses-q01","kind":"action"},{"text":"If the generators or measures also vary, what separate convergence result justifies the claimed limit?","id":"convergence-hypotheses-q02","kind":"boundary"}]}]},{"id":"integration-by-parts","name":"Integration by parts","description":"Control boundary terms and simultaneous jumps in product-based transformations.","findings":[{"id":"jump-aware-product-rule","name":"Jump-aware product rule","description":"Record regularity assumptions, one-sided evaluations and any required jump correction.","questions":[{"text":"Which integration-by-parts identity applies to the stated regularity and endpoint conventions?","id":"jump-aware-product-rule-q01","kind":"action"},{"text":"Do the functions share jumps, and how does the selected identity account for their jump products?","id":"jump-aware-product-rule-q02","kind":"boundary"}]}]}]},{"id":"interpretations-and-neighbours","name":"Interpretations and neighbours","description":"Connect the construction to probability and related integration theories without conflating their assumptions.","layers":[{"id":"probability-interpretation","name":"Probability interpretation","description":"Identify when the generator is a cumulative distribution function and the integral represents an expectation.","findings":[{"id":"expectation-against-a-law","name":"Expectation against a law","description":"Link a distribution function F_X to the law of X and the corresponding integral of g.","questions":[{"text":"Does F satisfy the monotonicity, right-continuity and limits at infinity required of a cumulative distribution function?","id":"expectation-against-a-law-q01","kind":"definition"},{"text":"Under what integrability conditions does the proposed expectation E[g(X)] equal ∫g(x) dF_X(x)?","id":"expectation-against-a-law-q02","kind":"action"}]}]},{"id":"comparison-with-riemann-stieltjes","name":"Comparison with Riemann-Stieltjes integration","description":"Identify the definition in use and justify any claimed agreement between theories.","findings":[{"id":"definition-and-agreement","name":"Definition and agreement","description":"Distinguish a measure-based integral from a partition-sum definition and document conditions for coincidence.","questions":[{"text":"Does the cited treatment define the integral through a generated measure or through tagged partition sums?","id":"definition-and-agreement-q01","kind":"provenance"},{"text":"Which hypotheses guarantee agreement for this integrand and generator, and are their endpoint conventions compatible?","id":"definition-and-agreement-q02","kind":"boundary"}]}]}]},{"id":"evidence-and-external-alignment","name":"Evidence and external alignment","description":"What the world already says about this thing, gathered so the model can be checked against it.","layers":[{"id":"reported-evidence","name":"Reported evidence","description":"Findings from the breadth pass, kept separate from the structural claims.","findings":[{"id":"evidence-confidence-notes","name":"Check these first","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these check these first hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","kind":"provenance"}]},{"id":"evidence-kinds","name":"Kinds and varieties","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these kinds and varieties hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?","id":"evidence-kinds-q01","kind":"provenance"}]},{"id":"evidence-real-world-use","name":"Real-world use","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these real-world use hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?","id":"evidence-real-world-use-q01","kind":"provenance"}]},{"id":"evidence-failure-modes-and-hazards","name":"Failure modes and hazards","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","kind":"provenance"}]},{"id":"evidence-neighbours","name":"Neighbouring kinds and how to tell them apart","description":"Recalled without web access and unsourced; every item is a lead to verify.","questions":[{"text":"Which of these neighbouring kinds and how to tell them apart hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Explain the theory with correct definitions, stating its assumptions.","Check computations and proofs, showing each step."],"mustNot":["Present results that need conditions, such as bounded variation, as holding without them.","Attribute theorems to the wrong mathematicians.","Present a faulty proof as complete.","Help students cheat in graded work."],"requiresHuman":["Using results in financial or engineering models that affect people."]},"ethics":{"considerations":["Mathematical errors in applied models can mislead decisions about money and risk.","Honest teaching respects learners' own work."],"affectedParties":["Students","Researchers","People affected by applied models"]},"owners":{"steward":"Nobody owns mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q5638607","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.lebesgue-stieltjes-integration","Wikidata: Q3799278 (https://www.wikidata.org/wiki/Q3799278)"]},"properties":{"applicability":"not-applicable","items":["Generating-function class: Nondecreasing real-valued; locally bounded variation; bounded variation; other explicitly specified extension","Continuity and interval convention: Right-continuous with (a,b]; left-continuous with [a,b); explicitly documented alternative","Associated measure: Generator linked to a Borel measure and, where used, its completion","Integration domain: Specified Borel set; interval with explicit endpoints; whole real line","Integral status: Not yet established; finite; positive infinity; negative infinity; undefined because both positive and negative parts diverge; inadmissible under the stated framework","Absolute-integrability quantity: ∫|f| dμ_F for positive measures, or ∫|f| d|μ_F| for signed or complex measures; value in [0,∞]","Measure-component profile: Atomic; absolutely continuous; singular continuous; mixture","Generator equivalence: Generators differing by an additive constant under the same interval convention generate the same measure"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["Construct or identify the measure generated by an admissible integrator.","Check whether an integrand is measurable and whether its integral is finite, extended or undefined.","Evaluate contributions from point masses and densities while retaining any singular continuous remainder.","Translate a probability expectation into an integral against a cumulative distribution function.","Approve or reject limit exchanges and integration-by-parts steps by checking their hypotheses.","Compare alternative endpoint conventions or Riemann-Stieltjes formulations and explain any discrepancy."]},"hazards":{"applicability":"required","items":["Misapplied theorems in probability and finance models."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["Lebesgue-Stieltjes integration is used in measure theory, probability and analysis to integrate with respect to measures generated by suitable functions.","In probability, it expresses expectations using a distribution function while accommodating discrete, continuous and mixed distributions.","Its use requires explicit conventions for the generating function and interval endpoints when jumps contribute mass."]}},"sources":[{"title":"Wikidata item Q3799278: Lebesgue–Stieltjes integration","url":"https://www.wikidata.org/wiki/Q3799278","note":"identity and sense of the item"},{"title":"Wikipedia: Lebesgue–Stieltjes integration","url":"https://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration","note":"general description of the item"}],"openQuestions":["Which authoritative references should establish this registry entry's definition, notation and relationship to neighbouring integration concepts?","Should signed and complex bounded-variation integrators be fully owned by this entry or represented as linked extensions?","Which endpoint convention should Vercy adopt as its default, and how should alternative conventions be recorded?","Should locally bounded-variation generators on unbounded domains be supported directly, and which global integrability framework should govern them?","Which existing world model, if any, already covers this concept and should remain its single source of truth?"],"resources":{"spec":"/models/things/publications/thing-q3799278/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q3799278/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T10:55:11Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"not-applicable","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T10:10:30+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No governing technical standard, law or official classification is required for this mathematical construction.","interaction.recognition":"This abstract mathematical construction has no physical appearance that uniquely identifies it."},"score":1.0}}