APPLIED MATHEMATICS
Mathematical structure for computational engineering.
Notes on analysis, integration, function spaces, approximation, and the ideas that make numerical and geometric systems understandable.
CURATED SERIES
The Lebesgue–Stieltjes Integral
A rigorous chapter-by-chapter study of Carter and van Brunt: from the completeness of the real line to measure construction, convergence theorems, Lp spaces, Hilbert geometry, and gauge integration.
- 01 Real Numbers
- 02 Analytic Preliminaries
- 03 The Riemann Integral
- 04 The Lebesgue–Stieltjes Integral
- 05 Core Properties and Convergence
- 06 Integral Calculus
- 07 Product Measures and Repeated Integrals
- 08 Lebesgue Spaces Lᵖ
- 09 Hilbert Spaces and L²
- 10 Beyond Lebesgue Integration
EDITORIAL STANDARD
Definitions, hypotheses, derivations, and failure cases.
Precise conventions
Every article fixes its measure, interval convention, function space, and convergence mode before using a theorem.
Qualified theorems
Results state the necessary measurability, integrability, monotonicity, continuity, and finiteness hypotheses.
Counterexamples
Boundary cases show exactly where an interchange of limits, integrals, or summation order ceases to be valid.
Computational meaning
Each chapter connects the analytical result to approximation, geometry processing, numerical integration, or simulation.
ENGINEERING CONNECTION
Why this mathematics matters.
Approximation
Projection, bases, norms, and convergence clarify what an algorithm preserves and how error should be measured.
Numerical methods
Integration and function spaces give finite computations a rigorous relationship to continuous problems.
Geometry processing
Continuity, parameterization, measure, and transformations shape reliable geometric representations.
Simulation
Hilbert-space geometry and variational ideas underlie least squares, spectral methods, and many engineering solvers.
PRIMARY REFERENCES