# Concept derivation guides

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Concept guide



## [Measurable function: checking every preimage by complete enumeration, with composition, indicators and the image-set trap](/en/derivations/derivation-mko-measurable-function-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-measurable-function)

X={0,1,2,3}, Σ_X={∅,{0,1},{2,3},X}; Y={0,1,2}, Σ_Y={∅,{0},{1,2},Y}; Z={0,1} with Σ_Z the power set. (1) Prove by complete enumeration that f=(0↦0,1↦0,2↦1,3↦2) is measurable and g=(0↦0,1↦1,2↦1,3↦2) is not; (2) prove that a composite of measurable maps is measurable, and read φ∘f back as an indicator function; (3) prove that an indicator 1_E is measurable exactly when E∈Σ_X; (4) use m and k to show that 'checking images' can both wrongly reject and wrongly accept.

Measurable function

derivation-mko-measurable-function-01





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