Simple Function - Relation To Lebesgue Integration

Relation To Lebesgue Integration

Any non-negative measurable function is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. Indeed, let be a non-negative measurable function defined over the measure space as before. For each, subdivide the range of into intervals, of which have length . For each, set

for, and .

(Note that, for fixed, the sets are disjoint and cover the non-negative real line.)

Now define the measurable sets

for .

Then the increasing sequence of simple functions

converges pointwise to as . Note that, when is bounded, the convergence is uniform. This approximation of by simple functions (which are easily integrable) allows us to define an integral itself; see the article on Lebesgue integration for more details.