This in turn led to important generalizations of the concept of integral designed to integrate highly discontinuous functions the Riemann integral of 1854 and the Lebesgue integral of 1902.
Then (f(x)) is improper fuzzy Riemann integrable on ([a,infty[) and the improper fuzzy Riemann integral is a fuzzy number. Furthermore, we have int_{a}^{infty}f(x),dx = biggl int _{a}^{infty}underline{f} x,r),dx, int _{a}^{infty}overline{f} x,r),dx biggr). (see [14]).
A real function must be integrated between limits a and b, and the Riemann integral is defined in terms of a sum involving values spread along the interval from a to b.
If the family ( ( S z α ) α ∈ Λ < b is not summable, then g is not HK integrable on [ a, ∞ ], whence the improper Riemann integral over [ a, b ] does not exist.
(The Riemann integral is explained in the section Integration).
(See the sections Riemann integral and Measure theory).
This done, the definition of the Riemann integral can be extended to the complex case.
His 1996 paper "Return to the Riemann Integral" earned one of the Mathematical Association of America's highest honors in 1997.
Once the measure is established, Lebesgue's generalization of the Riemann integral can be defined, and it turns out to be far superior to Riemann's integral.
However, the resulting function g now takes the form g(x) = 1 for all x, and this function does possess a Riemann integral.
If both of these sums converge to the same limiting value as the thickness of the slices tends to zero, then their common value is defined to be the Riemann integral of f between the limits a and b.
