A First Look at Rigorous Probability Theory

pg. 26

A collection \(\mathcal{J}\) of subsets of \(\Omega\) is a semialgebra if:

1. it contains \(\Omega\) and \(\phi\),
2. is closed under finite intersection,
3. complement of any element of \(\mathcal{J}\) is equal to finite disjoint union of elements of \(\mathcal{J}\)

pg. 27

1. it contains \(\Omega\) and \(\phi\),
2. closed under complement
3. is closed under finite unions and intersection,

So, a semialgebra closed under finite union and complement is a algebra.

pg. 24

A collection (\(\Sigma\)) of subsets of X is a \(\sigma\) - algebra if:

1. it contains X
2. closed under complement
3. closed under countable unions
4. closed under countable intersections (follows from 2 and 3)

So, a algebra closed under countable unions is a sigma algebra.

The ordered pair \((X, \Sigma)\) is called a measurable space.

Measure is defined only for \(\sigma\) - algebra. (Wikipedia)

• An example of Sigma algebra: A First Look at Rigorous Probability Theory - Jeffrey S. Rosenthal - 2006.pdf: Page 24
• Example of non-measurable set: A First Look at Rigorous Probability Theory - Jeffrey S. Rosenthal - 2006.pdf: Page 21

pg 24

A probability triple or measure space is a triple \((\Omega, \mathcal{F}, P)\), where:

• \(\Omega\) sample space is non empty set
• \(\mathcal{F}\) is \(\sigma\) - algebra of \(\Omega\)
• \(P\) is probability measure: mapping from \(\mathcal{F}\) to \([0,1]\) with \(P(\phi) = 0\) and \(P(\Omega) = 1\)

pg. 33

complete extension: if \(A \in \mathcal{M}\) with \(P^*(A) = 0\) and if \(B \subseteq A\) , then \(B \in \mathcal{M}\)

pg. 33

The smallest \(\sigma\) algebra containing \(\mathcal{J}\), \(\mathcal{B} = \sigma(\mathcal{J})\) is called the Borel \(\sigma\) - algebra of \(\mathcal {J}\) if it