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Lecture 02 - Jan 7

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From last class, Lecture 1 But there exists no such mapping.

Vitali’s construction

non-measurable set.

On , we define an equivalence relation if

Let

Example: is uncountable.

Axiom of Choice

Every indexed family of non-empty sets, there exists an indexed set such that for every . Given an equivalence class there exists

Define (well defined subset of ).

  1. if

Suppose

then for some ( - different equivalence classes)

The intersection is empty.

NOTE

Union of is large enough to cover the interval but small enough to fit within .

Every must have same measure. Take . Then for some .

i.e. .

If with .

Now, .

For any

As

Throughout this lecture, .

Definition (Semi-algebra)

is called a semi-algebra if:

  2. (closed under intersection)
  3. If then, such that disjoint

Exercise:

Definition (Algebra)

is called an Algebra if:

  2. (closed under union)
  3. (closed under complement)

NOTE

Algebra is closed under complements.

A single set in the collection semi-algebra only requires the complement to be a union of several sets from the collection.

An algebra is closed under intersection.

Definition (-Algebra)

is called a -Algebra if:

  2. (closed under complement)
  3. If (closed under union)

Exercise

  1. If is a family of algebras, then is also an algebra.

Contains Since each is an algebra, for all . Thus, . Closed under complement If , then for all . Since each is an algebra, for all . Thus, . Closed under finite union If , then for all . Since each is an algebra, for all . Thus, .

  1. If is a family of -Algebras, then is also a -algebra.

Contains For every , because each is a -algebra. Thus, . Closed under complement If , then for all . Since each is a -algebra, for all . Thus, . Closed under countable union Let . Then for each , for all . Since each is a -algebra, for all . Thus, .

Definition

  1. If , where is a class of subsets of , then

is called the algebra generated by .

NOTE

It is the intersection of all algebras on the set that contains .

Example

Let be uncountable. Consider

then is a -algebra.

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