Przykłady zagadnień z różnych działów matematyki niezbędnych do studiowania teorii miary

Abstract

In the education of mathematicians, including teachers of mathematics, the measure theory plays an important role. From the experience in and research on teaching the measure theory it follows that one of the important reasons why students encounter difficulties in the subject is their insufficient ability to apply their knowledge of other branches of mathematics, especially of the set theory and topology. In this article we propose a series of problem analyses and exercises aimed at preparing students to study the measure theory, especially to understand the proofs of theorems on properties of the measure, including the Lebesgue measure. Most of these problems and exercises are presented with solutions, outlines of solutions, hints, didactic remarks and comments. In this paper we point out, in a practical way, the significance of active reading of mathematical texts and skilful use of mathematical literature. This article is dedicated to both students of mathematics and their academic teachers.