These elementary functions and their elementary integrals may be any set of functions and definitions of integrals over these functions which satisfy these axioms.
3.
Defining the elementary integral of the family of step functions as the ( signed ) area underneath a step function evidently satisfies the given axioms for an elementary integral.
4.
Defining the elementary integral of the family of step functions as the ( signed ) area underneath a step function evidently satisfies the given axioms for an elementary integral.
5.
Using the family of all continuous functions as the elementary functions and the traditional Riemann integral as the elementary integral is also possible, however, this will yield an integral that is also equivalent to Lebesgue's definition.
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