Example 7.4.4(d): Lebesgue Integral for Simple Functions

Define two simple functions
s₁(x) = 2 X_{[0, 2]}(x) + 4 X_{[1, 3]}(x)
s₂(x) = 2 X_{[0, 1)}(x) + 6 X_{[1, 2]}(x) + 4 X_{(2, 3]}(x)
Show that s₁(x) = s₂(x) and s₁(x) dx = s₂(x) dx.
Context

To show that s₁(x) = s₂(x) is easy:

Take x [0, 1): Then s₁(x) = 2 and s₂(x) = 2.
Take x [1, 2]: Then s₁(x) = 2 + 4 = 6 and s₂(x) = 6.
Take x (2, 3]: Then s₁(x) = 4 and s₂(x) = 4.

Thus the two functions agree.

By definition we have:

s₁(x) dx = 2 m([0, 2]) + 4 m([1, 3]) = 4 + 8 = 12

and

s₂(x) dx = 2 m([0, 1)) + 6 m([1, 2]) + 4 m((2, 3]) = 2 + 6 + 4 = 12

so that the value of the integrals agree as well.

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

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007