Answered: Prove if f is a nonnegative measurable…

Prove if f is a nonnegative measurable function, then there exists an increasing sequence (ϕn) of simple functions that converges pointwise to f.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Simple Approximation Theorem: An extended real-value function f on a measurable set E is measurable if and only if there is a sequence {ϕ_n} of simple functon on E which converges pintwise on E to f and has the property that

|ϕ_n| ≤ |f| on E for all n.

If f is nonnegative, we may choose {ϕ_n} to be increasing.

Prove if f is a nonnegative measurable function, then there exists an increasing sequence (ϕ_n) of simple functions that converges pointwise to f.

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