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Upper Function

  • Definition: A real-valued function f defined on an interval I is called an upper function on I, and we write f is an element of U(I), if there exists an increasing sequence of step functions {s_n} such that a) s_n is an increasing sequence converging to f almost everywhere on I, and b) the limit as n goes to infinity of int(s_n) over I is finite. The sequence {s_n} is said to generate f. The integral of f over I is defined by the equation int_I(f) = lim_(n goes to infinity)[int_I(s_n)].

    Source: Mathematical Analysis, second edition by Tom M. Apostol

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