Lebesgue Integral in Constructive Analysis

Abstract

In the present note it is assumed that the reader is acquainted with [1]. We shall understand algorithms to be normal algorithms in the alphabet of constructive real numbers (CRN), extended by the letters Δ, □, β. Let n be a natural number, n ≥ 1. We shall call n-points words of the form x₁□x₂□...□x_n, where all the x_i (1 ≤ i ≤ n) are CRN from 0 Δ 1 and we shall call n-segments, words of the form

$$x_1^1 \vartriangle \,x_1^2 \,\square \,x_2^1 \,\vartriangle \,x_2^2 \,\square \cdots \square \,\,x_n^1 \vartriangle \,x_n^2 ,$$

((1))

where x ¹_j , x ²_j are CRN from 0Δ1, such that x ¹_j < x ²_j (1 ≤j ≤ n).