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The Gauge Integral, or Generalized Riemann Integral, was introduced in the sixties. It was the first successful attempt to define an integral which weakens the conditions on the classical Fundamental Theorem of Calculus more simply than the Lebesgue Integral while retaining a large body of integrable functions.

The Generalized Riemann Integral is, as the name suggests, an immediate generalization of the classical Riemann Integral. It can also be viewed as a generalization of the Lebesgue Integral after a clever manipulation of its definition.