Question

Please help me in understanding Generated Sigma Algebra:
Consider S = {(H,H), (H,T), (T,H), (T,T)},
the repeated coin toss. The Sigma-Algebra generated by
C = {{(HH), (TT)}} is
σ(C) = { ∅ , S, {(HT), (TH)}, {(HH), (TT)} }

How does "(H,T), (T,H)" jumped in the σ(C) ? What is the difference between σ-algebra & generated σ-algebra? Why "(HT), (TH)" in σ(C) have been seperated from "(HH), (TT)" with curly brackets?

Answer 1

@AccessDenied please help me

Answer 2

https://www.math.lsu.edu/~sengupta/7312s02/sigmaalg.pdf

Answer 3

And here's your other question: http://www.physicsforums.com/showthread.php?s=2e24a947927ada649126b86bb97997b6&p=3984544#post3984544

Answer 4

To directly answer your first question, {(HT),(TH)} is part of the sigma-algebra because it is the complement of C in S.

Answer 5

And they are separated with curly brackets because the objects in the sigma-algebra are sets themselves.

Answer 6

A generated sigma-algebra is basically just the smallest sigma-algebra that contains the generating set. Going off of the definition of sigma-algebra, you have to have the null set, the entire set S, the subset C, and C complement, which are the four elements you see there.

Answer 7

Does that make sense or do you have more questions about sigma-algebras?

Answer 8

@nbouscal thanks a tonne, actually I have posted the same question on physicsforums :) but you have explained very well