Researching on the approximation of fuzzy system to integrable function class by means of the integral norm(a metric) is one method of common concern to the people.The

Kp

-integral norm induced by the

K

-quasi-arithmetic operations is not only a generalization for a one dimensional integral norm

but also an important tool to describe the

p

-integrable function classes.In this paper

the

Kp

-integral norm is redefined by introducing the quasi-subtraction operator.In the sense of the

Kp

-integral norm

the approximation of the piecewise linear functions to a kind of

p

-integrable functions is discussed.Then

we prove constructively that the generalized Mamdani fuzzy system has the approximation to a class of

p

-integrable functions.Finally

by a practical example the approximation effect of the generalized Mamdani fuzzy systems is illustrated.The results show that the generalized Mamdani fuzzy system can approximate a kind of

p

-integrable functions to arbitrary accuracy.