Prelim Study Guide

Definitions

  • ℬ(ℝ)\mathcal{B}(\mathbb{R})
  • countable/co-countable σ\sigma-field
  • field
  • σ\sigma-field generated by 𝒞\mathcal{C}
  • lim inf⁡\liminf and lim sup⁡\limsup and lim⁡\lim
  • non-decreasing, non-increasing, and monotone sequences
  • σ\sigma-field
  • distribution function
  • general construction of probability model
  • λ\lambda-system
  • π\pi-system
  • probability space
  • semialgebra
  • measure induced by random element
  • measurable map
  • measurable space
  • preimage

    • Theorems

  • minimal postulates for a field 𝒜\mathcal{A}
  • properties of indicators
  • intuitive interpretation of lim inf⁡ \liminf and lim sup⁡ \limsup
  • properties of lim inf⁡ \liminf and lim sup⁡ \limsup
  • properties of monotone sequences
  • minimal postulates for a σ\sigma-field 𝒜\mathcal{A}
  • key properties of σ\sigma-fields
  • restriction of σ\sigma-fields
  • combo extension
  • Dynkin
  • characterization of the field generated by a semialgebra 𝒮\mathcal{S}
  • first measure extension
  • the eight properties of probability measures
  • second measure extension
  • measurability of composition
  • necessary sufficient conditions for measurability
  • properties of preimages
  • preimages of σ\sigma-fields