extbf {Math136 / Stat219 Course Goals} extbf {Why Measure-Theoretic Probability?} extbf {Measurable space} extbf {Choosing the $sigma $-field $clf $} extbf {Generated $sigma $-field} extbf {Borel $sigma $-field} extbf {Example: showing two $sigma $-fields are equal}Math136 / Stat219 Course Goals•Basic concepts and definitions of measure-theoreticprobability and stochastic processes•Properties of key stochastic processes and theirapplications, especially Brownian motion•Key results and common techniques of proof•Preparation for further study (especially for Math 236:stochastic differential equations)Today’s lecture: Sections 1.1MATH136/STAT219 Lecture 1, September 22, 2008 – p. 1/7Why Measure-Theoretic Probability?•Mathematical models of physical processes•Outcome is uncertain or “random”•Probability = “Language”•Measure Theory = “Grammar”•Measure theory allows us to consider◦General random variables◦Arbitrary probability spacesMATH136/STAT219 Lecture 1, September 22, 2008 – p. 2/7Measurable space•ω: outcome of random experiment•Ω: sample space - set of all possible outcomes•A collection, F, of subsets of Ω is a σ-field (aka σ-algebra)if:◦Ω ∈ F◦if A ∈ F then Ac∈ F◦if Ai∈ F, i = 1, 2, . . . thenS∞i=1Ai∈ F•A measurable space is a pair (Ω, F) is a σ-field of subsetsof Ω•F: event space - all events of interest; all events we willassign probabilities toMATH136/STAT219 Lecture 1, September 22, 2008 – p. 3/7Choosing the σ-field F•Countable sample spaceF = 2Ω•Uncountable sample space◦2Ωis too big (for an example, see Rosenthal Section 1.2)◦Require inclusion of certain “nice" sets◦Take F to be smallest σ-field that includes these setsMATH136/STAT219 Lecture 1, September 22, 2008 – p. 4/7Generated σ-field•Let Γ be an arbitrary index set, and let {Aα: α ∈ Γ} be acollection of subsets of Ω•The σ-field generated by {Aα: α ∈ Γ}, denotedσ({Aα: α ∈ Γ}), is the smallest σ-field that contains thecollection {Aα: α ∈ Γ}•σ({Aα: α ∈ Γ}) is the intersection of all σ-fields that contain{Aα: α ∈ Γ}•Note: intersection of an arbitrary collection of σ-fields is aσ-field◦But an arbitrary union of σ-fields is not necessarily aσ-fieldMATH136/STAT219 Lecture 1, September 22, 2008 – p. 5/7Borel σ-field•The Borel σ-field on IR is B.= σ({(a, b) : a < b ∈ IR})•Some equivalent definitions:B = σ({[a, b] : a < b ∈ IR}) = σ({(a, b] : a < b ∈ IR})= σ({(a, b) : a < b ∈ Q}) = σ({[a, b] : a < b ∈ Q})= σ({(−∞, b] : b ∈ IR}) = σ({(−∞, b] : b ∈ Q})= σ({open sets of IR})•The Borel σ-field on IRnisBn.= σ({(a1, b1) × · · · × (an, bn) : ai, bi∈ IR, i = 1, . . . , n})•In general, the Borel σ-field on a space Ω is defined asσ({open sets of Ω})MATH136/STAT219 Lecture 1, September 22, 2008 – p. 6/7Example: showing two σ-fields are equal•Consider the σ-fields◦B = σ({(a, b) : a < b ∈ IR})◦G = σ({(a, b], a < b ∈ IR})•(a, b) ∈ G for all a < b ∈ IR ⇒ B ⊆ G◦{b} = ∩∞n=1(b −1n, b] ∈ G◦(a, b) = (a, b] \ {b} ∈ G•(a, b] ∈ B for all a < b ∈ IR ⇒ G ⊆ B◦{b} = ∩∞n=1(b −1n, b +1n) ∈ B◦(a, b] = (a, b) ∪ {b} ∈ BMATH136/STAT219 Lecture 1, September 22, 2008 – p.