Learn Construct Riemann–Stieltjes sums, prove existence under explicit hypotheses, and distinguish refinement convergence from arbitrary fine-mesh convergence.
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Weighted sums reuse containment of extrema but replace widths with integrator increments.
Each child cell has a higher infimum and lower supremum; its nonnegative weight contributes to a sum whose weights add to the parent weight.
The point value determines which adjacent cell receives each part of the increment.
It only places that value between the two one-sided limits. Different allowed values can redistribute the jump weight.
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Mixed weights, jump values and existence boundaries
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Build weighted Darboux integration from nonnegative increments, prove its refinement criterion, and state exactly how arbitrary tagged sums converge.
6 practice tasks.
Separate refinement integrability from all-small-mesh tagged convergence and account for the exact mass carried by interior and endpoint jumps.
6 practice tasks.
Control weighted gaps and every sufficiently fine tagged sum, including finite discontinuities away from integrator jumps.
6 practice tasks.
Use a derivative density only when justified, define signed smooth integrators, and derive boundary terms from sums.
6 practice tasks.