Lebesgue–Stieltjes integration

Monotonicity and variation

Record whether F generates a positive measure or requires a qualified signed or complex extension.

1. Is F finite-valued and nondecreasing, or is the construction based on bounded variation? definition
2. Are the required variation and finiteness assumptions local to the integration interval or valid on the entire domain? boundary

Interval increments

Make the correspondence between increments of F and interval masses explicit.

1. Does the construction use right-continuous F with μ_F((a,b]) = F(b) − F(a), or another stated convention? definition
2. Which reference fixes this convention, and is integration performed on the Borel σ-algebra or its completion? provenance

Measure-relative integrand

Assess measurability and equality almost everywhere with respect to the associated measure.

1. Is f measurable for the chosen measurable space and measure completion? definition
2. Would changing f on a Lebesgue-null set alter its values on a set with positive μ_F mass? boundary

Positive, negative and absolute parts

Use the appropriate positive-part or total-variation tests for the selected measure framework.

1. For a positive μ_F and real f, what are ∫f⁺ dμ_F and ∫f⁻ dμ_F, and do both diverge? measurement
2. For a finite signed or complex measure, is ∫|f| d|μ_F| finite? measurement

Jump contributions

Under the right-continuous convention, an included atom x contributes f(x)[F(x) − F(x−)].

1. Which included points have nonzero jump mass, and what values of f are assigned there? measurement
2. Does the notation ∫ from a to b denote integration over (a,b], [a,b], [a,b) or another explicitly defined set? boundary

Density and singular remainder

Record the absolutely continuous contribution and any singular continuous remainder.

1. What is the density of the absolutely continuous component with respect to Lebesgue measure? measurement
2. What evidence establishes that a singular continuous component is absent or must be retained? boundary

Convergence hypotheses

Attach the relevant theorem and verified assumptions to a proposed interchange.

1. For a fixed positive μ_F, do the integrands satisfy monotone convergence or dominated convergence hypotheses? action
2. If the generators or measures also vary, what separate convergence result justifies the claimed limit? boundary

Jump-aware product rule

Record regularity assumptions, one-sided evaluations and any required jump correction.

1. Which integration-by-parts identity applies to the stated regularity and endpoint conventions? action
2. Do the functions share jumps, and how does the selected identity account for their jump products? boundary

Expectation against a law

Link a distribution function F_X to the law of X and the corresponding integral of g.

1. Does F satisfy the monotonicity, right-continuity and limits at infinity required of a cumulative distribution function? definition
2. Under what integrability conditions does the proposed expectation E[g(X)] equal ∫g(x) dF_X(x)? action

Definition and agreement

Distinguish a measure-based integral from a partition-sum definition and document conditions for coincidence.

1. Does the cited treatment define the integral through a generated measure or through tagged partition sums? provenance
2. Which hypotheses guarantee agreement for this integrand and generator, and are their endpoint conventions compatible? boundary

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence? provenance

Kinds and varieties

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1. Which of these kinds and varieties hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Lebesgue-Stieltjes integration this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. 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? provenance