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.
Read more about this topic: Simple Function
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