Notes on Measure and Integration

Lecture notes about a simpler approach to Riemann integration

It is of course well known that the usual definitions of Riemann integration and Riemann integrals are equivalent to simpler definitions which can be expressed in terms of just one sequence of partitions, using dyadic intervals or dyadic squares or dyadic cubes for univariate, double, or triple integrals respectively. These lecture notes, intended mainly for undergraduates, present and prove some basic standard results of Riemann integration in detail, taking advantage of this simpler definition. The last section of the notes provides a proof of the equivalence of this definition with the clas…

Ralph Henstock's Lectures on the Theory of Integration

These are the class notes of lectures given by Ralph Henstock at the New University of Ulster in 1970-71. The notes deal with the Riemann-complete integral (also known as the generalized Riemann integral, the gauge integral, and the Henstock-Kurzweil integral). They also introduce Henstock's abstract theory of integration.

Fractional integration for Laguerre expansions

The aim of this note is to provide a fractional integration theorem in the framework of Laguerre expansions. The method of proof consists of establishing an asymptotic estimate for the involved kernel and then applying a method of Hedberg \cite{pro}. We combine this result with sufficient $(p,p)$ multiplier criteria of Stempak and Trebels \cite{ST}. The resulting sufficient $(p,q)$ multiplier criteria are comparable with necessary ones of Gasper and Trebels \cite{laguerre}.

Notes on metrics, measures, and dimensions

These notes deal with metric spaces, Hausdorff measures and dimensions, Lipschitz mappings, and related topics. The reader is assumed to have some familiarity with basic analysis, which is also reviewed.

关于测度上基于傅里叶的度量完备性的一个注记

A note on the completeness of Fourier-based metrics on measures

本文将基于傅里叶的度量从概率测度推广到复Borel测度，并研究其完备性，扩展了Stawiska对Carillo-Toscani未决问题的解答。

Note on the Equilibrium Measures of Julia sets of Exceptional Jacobi Polynomials

We prove that similarly to the standard case, the equilibrium measure of Julia sets of exceptional Jacobi polynomials tends to the equilibrium measure of the interval of orthogonality in weak-star sense.