A Definition of Integrals

The Riemann integral can be generalized in various ways by modifying the construction and selection of Riemann sums. This paper is a variation based on the following observation. If a function ƒ on a space X is integrable, then the Riemann sums for ƒ should not be changed substantially if we ignore the points in a sufficiently small subset S of X. Once this is allowed, we can for example get rid of the points where ƒ takes extremely large values or behaves too wildly. This idea is used to define a notion of integral, and some examples are investigated.