Integrability of Bounded Total Functions

For simplicity, we use the following convention: i, n denote natural numbers, a, r, x, y denote real numbers, A denotes a closed-interval subset of R, C denotes a non empty set, and X denotes a set. We now state several propositions: (1) For every element D of divsA such that vol(A) = 0 holds lenD = 1. (2) χA,A is integrable on A and integralχA,A = vol(A). (3) For every partial function f from A to R and for every r holds f is total and rng f = {r} iff f = r χA,A. (4) Let f be a partial function from A to R and given r. If f is total and rng f = {r}, then f is integrable on A and integral f = r · vol(A). (5) For every r there exists a partial function f from A to R such that f is total and rng f = {r} and f is bounded on A. (6) Let f be a partial function from A to R and D be an element of divsA. If vol(A) = 0, then f is integrable on A and integral f = 0.