CourseInfo 1.1 Probability spaces, Lecture notes of Algebra

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Math 561, Spring 2022

Renming Song

University of Illinois at Urbana-Champaign

January 19, 2022

Outline

Course syllabus is available from my homepage: https://faculty.math.illinois.edu/∼rsong/561s22/561s222.html

Textbook: R. Durrett: Probability: Theory and Examples (5th edition) Cambridge University Press, 2019

You do need a copy of this book. Most of the homework assignment will be from this book.

Office Hours: MWF: noon-12:50 pm in 227 CAB. I will also be on Zoom during this time.

Course syllabus is available from my homepage: https://faculty.math.illinois.edu/∼rsong/561s22/561s222.html

Textbook: R. Durrett: Probability: Theory and Examples (5th edition) Cambridge University Press, 2019

You do need a copy of this book. Most of the homework assignment will be from this book.

Office Hours: MWF: noon-12:50 pm in 227 CAB. I will also be on Zoom during this time.

Course syllabus is available from my homepage: https://faculty.math.illinois.edu/∼rsong/561s22/561s222.html

Textbook: R. Durrett: Probability: Theory and Examples (5th edition) Cambridge University Press, 2019

You do need a copy of this book. Most of the homework assignment will be from this book.

Office Hours: MWF: noon-12:50 pm in 227 CAB. I will also be on Zoom during this time.

The first few lecture will be a quick review/summary of basic abstract measure theory. For those of you who are not familiar with abstract measure theory, you need to spend some extra time to catch up. We will use these in later chapters.

40% of your grade will depend on homework assignment, 30% will depend on the midterm test and 30% on the take-home final exam.

The Midterm test is on Friday, March 11 in our regular classroom during class time. The take-home final exam is due on May 6.

The first few lecture will be a quick review/summary of basic abstract measure theory. For those of you who are not familiar with abstract measure theory, you need to spend some extra time to catch up. We will use these in later chapters.

40% of your grade will depend on homework assignment, 30% will depend on the midterm test and 30% on the take-home final exam.

The Midterm test is on Friday, March 11 in our regular classroom during class time. The take-home final exam is due on May 6.

Outline

(^1) Course Info

(^2) 1.1 Probability spaces

Let Ω be a non-empty set. A collection F of subsets of Ω is called a σ-field (or σ-algebra) if (i) Ω ∈ F; (ii) A ∈ F ⇒ Ac^ ∈ F; (iii) Ai ∈ F, i = 1 , 2 , · · · ⇒ ∪∞ i= 1 Ai ∈ F.

If F is a σ-field of subsets of Ω, then (i) F is closed under finite union; (ii) F is closed under (finite or countable) intersection.

Given any collection C of subsets of Ω, there is a smallest σ-field F containing C, that is, there is a σ-field F containing C such that if F′^ is any σ-field F containing C, then F′^ ⊃ F.

The smallest σ-field F containing C is called the σ-field generated by C and usually denoted by σ(C).

A σ-field F is said to be countably generated if there is a countable subcollection C ⊂ F such that F = σ(C).

Given any collection C of subsets of Ω, there is a smallest σ-field F containing C, that is, there is a σ-field F containing C such that if F′^ is any σ-field F containing C, then F′^ ⊃ F.

The smallest σ-field F containing C is called the σ-field generated by C and usually denoted by σ(C).

A σ-field F is said to be countably generated if there is a countable subcollection C ⊂ F such that F = σ(C).

Example

Let Rd^ be the d-dim Euclidean space. The smallest σ-field containing all open sets of Rd^ is called the Borel σ-field on Rd^ and is denoted by Rd^. When d = 1, we simply write R.

Rd^ is countably generated since Rd^ is generated by

{(a 1 , b 1 ] × · · · × (ad , bd ] : ai < bi , ai , bi rational , i = 1 ,... , d}.

By a measurable space we mean a couple (Ω, F), where Ω is a non-empty set and F is a σ-field of subsets of Ω.

By a measure μ on a measurable space (Ω, F), we mean a [ 0 , ∞]-valued function μ on F such that (i) μ(∅) = 0; (ii) Ei ∈ F, i = 1 , 2 ,... , {Ei } disjoint ⇒

μ (∪∞ i= 1 Ei ) =

∑^ ∞

i= 1

μ(Ei ).

If μ(Ω) = 1, we say that μ is a probability measure. We usually denote a probability measure by P.

By a measurable space we mean a couple (Ω, F), where Ω is a non-empty set and F is a σ-field of subsets of Ω.

By a measure μ on a measurable space (Ω, F), we mean a [ 0 , ∞]-valued function μ on F such that (i) μ(∅) = 0; (ii) Ei ∈ F, i = 1 , 2 ,... , {Ei } disjoint ⇒

μ (∪∞ i= 1 Ei ) =

∑^ ∞

i= 1

μ(Ei ).

If μ(Ω) = 1, we say that μ is a probability measure. We usually denote a probability measure by P.