Note on integration of series by Lebesgue integrals

Notes on Lebesgue Integration

These notes record the lectures on Lebesgue integration, which is a topic not covered in the textbook. It is largely inspired by the approach of Leon Simon when he previously taught this course. (All mistakes are my own of course!) The notes will be posted after each lecture. Any comments or corrections, even very minor ones, are very much appreciated! Our goal is to define the notion of Lebesg...

The Riemann and Lebesgue Integrals

§1 Preliminaries: Step Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 §2 Riemann Integrable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 §3 Lebesgue measure zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 §4 Definition and Properties of the Lebesgue Integral . . . . . . . . . . 7 §5 The spaces L(R) and L(R) ....

A Numerical Integration Method by Using Generalized Series of Functions

On the Substitution Rule for Lebesgue–stieltjes Integrals

We show how two change-of-variables formulæ for Lebesgue–Stieltjes integrals generalize when all continuity hypotheses on the integrators are dropped. We find that a sort of “mass splitting phenomenon” arises. Let M : [a, b]→ R be increasing. Then the measure corresponding to M may be defined to be the unique Borel measure μ on [a, b] such that for each continuous function f : [a, b] → R, the i...

A Note on Complete Integrals

We present a theorem concerning the representation of solutions of a first-order partial differential equation in terms of a complete integral of the equation. We discuss the geometrical significance of that theorem.