Familiarizing Students with Definition of Lebesgue Outer Measure Using Mathematica: Some Examples of Calculation Directly from Its Definition

Abstract

In this paper we present some examples of calculation the Lebesgue outer measure of some subsets of mathbb {R}^2 directly from definition 1. We will consider the following subsets of mathbb {R}^2: displaystyle {(x, y)in mathbb {R}^2: 0le y le x^2, xin [0, 1]}, displaystyle {(x, y)in mathbb {R}^2: 0le y le exp (-x), xge 0}, big { (x,y) in mathbb {R}^2: ln x le y le 0, xin (0, 1]big }, big { (x,y) in mathbb {R}^2: 0le y le 1/x, xge 1big }, displaystyle {(x, y)in mathbb {R}^2: 0le y le sin x, xin [0, pi /2]}, displaystyle {(x, y)in mathbb {R}^2: 0le y le exp (x), xin [0, 1]}, displaystyle {(x, y)in mathbb {R}^2: 0le y le ln (1-2r cos x+r^2), x in [0, pi ]}, r>1 and some others. We could not find any analogical examples in available literature (except for rectangle and countable sets), so this paper is an attempt to fill this gap. We calculate sums, limits and plot graphs and dynamic plots of needed sets and unions of rectangles sums of which volumes approximate Lebesgue outer measure of the sets, using Mathematica. We also show how to calculate the needed sums and limits by hand (without CAS). The title of this paper is very similar to the title of author’s article (Wojas and Krupa in Math Comput Sci 11:363–381, 2017) which deals with definition of Lebesgue integral but this paper deals with definition of Lebesgue outer measure instead.