# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Sigma-algebras and measure spaces: an atom-generated sigma-algebra and a probability measure on a four-point carrier](/en/derivations/derivation-mko-sigma-algebra-measure-space-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-sigma-algebra-measure-space)

On X={0,1,2,3}, the 8 sets generated by the atoms A_1={0}, A_2={1,2}, A_3={3} form Σ, and μ(∪_{i∈I}A_i)=Σ_{i∈I}m_i with m_1=1/2, m_2=1/3, m_3=1/6. Prove that Σ is a sigma-algebra (with the finite reduction of countable unions), that μ is a measure (with the finite reduction of countable additivity) and a probability measure; then show that 𝒢={∅,{0},{1},X} is not a sigma-algebra and that ν, which changes a single value, is not a measure.

Sigma-algebras and measure spaces

derivation-mko-sigma-algebra-measure-space-01





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