Measure theory MOC

Measurable function

A measurable functions is a structure-preserving map of measurable spaces. Let (𝑋,Ξ£) and (π‘Œ,T) be measurable spaces. A function 𝑓 :𝑋 β†’π‘Œ is called measurable iff the preΓ―mage of every measurable set is measurable1 , #m/def/measure i.e. π‘“βˆ’1(𝐸) ∈Σ for any 𝐸 ∈T.

Properties

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

#state/tidy | #lang/en | #SemBr

Footnotes

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