A Riemann-Type Integral of Lebesgue Power

Abstract

The introduction of a mathematics student to formal integration theory usually follows the lines laid down by Riemann and Darboux. Later a change of ideas is necessary if he tackles Lebesgue's more powerful theory, and connections between the two are laboriously constructed. On the other hand, the commonest method of evaluating an integral is through the operation inverse to differentiation (the indefinite integral taken between limits). We refer to this as the calculus integral; few realize that this can succeed in cases where even the Lebesgue integral does not exist, let alone the Riemann one. An example is given later.