Measure theory MOC

Measurable function

A measurable functions is a structure-preserving map of measurable spaces. Let (𝑋,Σ) and (𝑌,T) be measurable spaces. A function 𝑓 :𝑋 →𝑌 is called measurable iff the preïmage of every measurable set is measurable¹ , measure i.e. 𝑓−1(𝐸) ∈Σ for any 𝐸 ∈T.

Properties

1. A measurable function from a measure space induces a Pushforward measure on its codomain.

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Footnotes

1. Note this is analogous to the topological definition of Continuity. ↩