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An integral of the Perron type McGregor, James Lewin

Abstract

In the definition of the Perron integral of a function f (x) over a closed interval [a, b] a major function M(x) and a minor function m(x) are required to satisfy the conditions (i) M(x) and m(x) are continuous on [a, b] and M(a) = m(a) = 0 ; (ii) - ∞ ≠ Ḏ M(x) ≥ f (x) ≥ D [overscored] m(x) ≠ + ∞. It is shown that without restricting the generality of the integral one may impose the additional condition (iii) M(x) and m(x) are differentiable on [a, b].

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