An introduction to Edalat’s theory of R-integration

1.1 The Riemann integral on the real line Let [a, b] be a compact interval of the real line and f : [a, b] → R be a (bounded) function. For the Riemann-integral of f over [a, b] one defines lower and upper (Darboux) sums with respect to a subdivision p of the form a = k0 < k1 < . . . < kn = b: S(f, p) := ∑n i=1(ki − ki−1) inf {f(x) | ki−1 ≤ x ≤ ki} S(f, p) := ∑n i=1(ki − ki−1) sup {f(x) | ki−1 ≤ x ≤ ki} These have a geometric interpretation as areas below, resp. around the given function