Abstract
Using an equivalence relation on the hyperreals called approximation, a new extension of the Riemann integral is motivated and introduced in which every bounded function is integrable and for which there exists a function g: [0, 1] → R simultaneously satisfying (1) g is integrable, (2) g is unbounded on every subinterval of [0, 1], and (3) g is identical to its average value function.
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