# SWARTHMORE MATH 136 - MATH 136 LECTURE 1 (7 pages)

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## MATH 136 LECTURE 1

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## MATH 136 LECTURE 1

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Pages:
7
School:
Swarthmore College
Course:
Math 136 - Stochastic Processes
##### Stochastic Processes Documents
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Math136 Stat219 Course Goals Basic concepts and definitions of measure theoretic probability and stochastic processes Properties of key stochastic processes and their applications 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 1 MATH136 STAT219 Lecture 1 September 22 2008 p 1 7 Why 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 spaces MATH136 STAT219 Lecture 1 September 22 2008 p 2 7 Measurable space outcome of random experiment sample space set of all possible outcomes A collection F of subsets of is a field aka algebra if F if A F then Ac F S if Ai F i 1 2 then i 1 Ai F A measurable space is a pair F is a field of subsets of F event space all events of interest all events we will assign probabilities to MATH136 STAT219 Lecture 1 September 22 2008 p 3 7 Choosing the field F Countable sample space F 2 Uncountable sample space 2 is too big for an example see Rosenthal Section 1 2 Require inclusion of certain nice sets Take F to be smallest field that includes these sets MATH136 STAT219 Lecture 1 September 22 2008 p 4 7 Generated field Let be an arbitrary index set and let A be a collection of subsets of The field generated by A denoted A is the smallest field that contains the collection A A is the intersection of all fields that contain A Note intersection of an arbitrary collection of fields is a field But an arbitrary union of fields is not necessarily a field MATH136 STAT219 Lecture 1 September 22 2008 p 5 7 Borel field The Borel field on IR is B a b a b IR Some equivalent definitions B a b a b IR a b a b IR a b a b Q a b a b Q b b IR b b Q open sets of IR The Borel field on IRn is Bn a1 b1 an bn ai bi IR i 1 n In general the Borel field on a space is defined as open sets of MATH136 STAT219 Lecture 1 September 22 2008 p 6 7 Example showing two fields are equal Consider the fields B a b a b IR G a b a b IR a b G for all a b IR B G 1 b b G b n 1 n a b a b b G a b B for all a b IR G B 1 1 b b B b n 1 n n a b a b b B MATH136 STAT219 Lecture 1 September 22 2008 p 7 7

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