## 7.4. Lebesgue Integral

### Example 7.4.2(a): Simple Functions

A

Suppose
**step function**is a function*s(x)*such that*s(x) = c*for_{j}*x*and the_{j-1}< x < x_{j}*{ x*form a partition of_{j}}*[a, b]*. Upper, Lower, and Riemann sums are examples of step functions. What is the difference, if any, between step functions and simple functions.*s(x) = c*for

_{j}*x*and the

_{j-1}< x < x_{j}*{ x*form a partition of

_{j}}*[a, b]*. Define functions

*X*such that

_{j}(x)*X*if

_{j}= 1*x*and

_{j-1}< x < x_{j}*0*otherwise. Then each

*X*is a characteristic function of the interval

_{j}*[x*and the sum

_{j-1}, x_{j}]is a simple function because (sub)intervals are measurable. ButS(x) = c_{j}X_{j}(x)

*S(x) = s(x)*, so that every step function is also a simple function.

But not every simple function is a step function. Take, for example, the set
* Q* of rational numbers inside

*[0, 1]*and

*. Then the function*

*= [2, 3]***A**is a simple function but not a step function.S(x) = X_{Q}(x) + X_{A}(x)

Therefore, simple functions are more general than step functions.