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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“In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
—Simone Weil (1909–1943)
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