We've set the stage for the Lebesgue integral in the previous two chapters; now it's time for the star to make her entrance. By way of a reminder, recall that we want our new integral to satisfy at least the following few, loosely stated properties:

• ∫ χE = m(E), whenever E is measurable.

• The integral should be linear: ∫(αf + βg) = α ∫ f + β ∫ g.

• The integral should be positive (or monotone): f ≥ 0 ⇒ ∫ f ≥ 0 (or f ≥ g ⇒ ∫ f ≥ ∫ g). In the presence of linearity, these are the same.

• The integral should be defined for a large class of functions, including at least the bounded Riemann integrable functions, and it should coincide with the Riemann integral whenever appropriate.

The first two properties tell us how to define the integral for simple functions. Once we know how to integrate simple functions, the third property suggests how to define the integral for nonnegative measurable functions: If f ≥ 0 is measurable, then we can find a sequence (ϕn) of simple functions that increase to f. Now set ∫ f = limn→∞ ∫ ϕn. Finally, linearity supplies the appropriate definition for the general case: If f is measurable, then f+ and f− are nonnegative, measurable, and f = f+ − f−.