MathCS.org - Real Analysis: Example 7.4.4(d): Lebesgue Integral for Simple Functions

7.4. Lebesgue Integral

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:

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.

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