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
Famous quotes containing the words relation to, relation and/or integration:
“To be a good enough parent one must be able to feel secure in one’s parenthood, and one’s relation to one’s child...The security of the parent about being a parent will eventually become the source of the child’s feeling secure about himself.”
—Bruno Bettelheim (20th century)
“To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
—Henry James (1843–1916)
“The more specific idea of evolution now reached is—a change from an indefinite, incoherent homogeneity to a definite, coherent heterogeneity, accompanying the dissipation of motion and integration of matter.”
—Herbert Spencer (1820–1903)