9.1 Product of two sigma-algebras

The product of two sigma-algebras is the sigma-algebra generated by the projection maps.

9.2 Product of many sigma-algebras

The product of many—even uncountably many—sigma-algebras is defined in the same way as the product of two sigma-algebras.

9.3 Borel x Borel = Borel

The product of two or more Borel sigma-algebras yields the Borel sigma-algebra of the product space.

9.4 Product measure

The product of two measures is defined using a nested integral.

9.5 Product measure is a measure

The product of two sigma-finite measures is the unique measure on the product sigma-algebra which naturally factorises for boxes.

9.6 Fubini–Tonelli theorem

The order of integration can be changed whenever the integrand is integrable with respect to the product measure (Fubini) or the integrand is positive (Tonelli).


Alternative reading material

  • [Jacod & Protter, Chapters 10 and 12]
  • [Williams, Chapter 8]

Last modified: Monday, 8 February 2021, 2:06 PM