## Example 7.4.4(d): Lebesgue Integral for Simple Functions

Define two simple functions

To show that Show thats_{1}(x) = 2 X_{[0, 2]}(x) + 4 X_{[1, 3]}(x)

s_{2}(x) = 2 X_{[0, 1)}(x) + 6 X_{[1, 2]}(x) + 4 X_{(2, 3]}(x)

*s*and_{1}(x) = s_{2}(x)*s*._{1}(x) dx = s_{2}(x) dx*s*is easy:

_{1}(x) = s_{2}(x)- Take
*x [0, 1)*: Then*s*and_{1}(x) = 2*s*._{2}(x) = 2 - Take
*x [1, 2]*: Then*s*and_{1}(x) = 2 + 4 = 6*s*._{2}(x) = 6 - Take
*x (2, 3]*: Then*s*and_{1}(x) = 4*s*._{2}(x) = 4

By definition we have:

ands_{1}(x) dx = 2 m([0, 2]) + 4 m([1, 3]) = 4 + 8 = 12

so that the value of the integrals agree as well.s_{2}(x) dx = 2 m([0, 1)) + 6 m([1, 2]) + 4 m((2, 3]) = 2 + 6 + 4 = 12

In other words, the value of the integral is independent of the representation of the simple functions in this example.