Sigma Algebra

What is sigma algebra ? A sigma Algebra of a Set X is a collection of subsets that is closed under complement, union of 2 or more subsets, or intersection of 2 or more subsets. Simple right ? NO? okay lets try this

Toss a coin 2 times. Simple enough, right ? 4 possible outcomes HH,HT,TH,TT. So, what do we know about what the outcome will be at the start of the experiment ? Well we know that each of the 4 outcomes is 14$\frac{1}{4}$. But we are not interested in Probabilities.

Let define

Ω=(HH,HT,TH,TT)$$$$

so we know that it is somewhere in this set. hence our Information I0is

I0=(ϕ,Ω)$$$$

Consider ϕ as if the experiment does not happen. Don't worry too much about it, Probability P(ϕ)=0 so it won't happen.

So, now if I ask you, what happens are the first toss ? Well we know that it can be a Heads H1=(HH,HT) or a Tail T1=(TH,TT). so our Information after$\mathit{\text{after}}$ the first Toss is

I1=(ϕ,H1,T1,Ω)$$$$

Okay so Time to take a breath, what do I mean by Information ? Well what I want to say is that, after the first toss, whatever is there in I1 has probability of 1 or 0. Another way, to say the same thing is, if someone asks you, hey, are you in this element of the set? You can say, Yes or No with certainty.

Lets assume the First toss was Heads. Now you I ask you, hey are you in ϕ? Well we know that P(ϕ)=0 so no, we are not in ϕ, infact we are never in ϕ(we kinda hate it! ).

Okay so how about H1 ? we the experiment is a 2 coin toss experiment, H1 has all the end points what lead to the first toss being heads, which we know for sure. So Yes, we are in H1. What about T1 ? Well since we know that everything  T1has starts with a tail in the first toss.. No, we are not in T1.

Finally Ω$\mathrm{\Omega }$ what about him ? Well, Ω has All the possible values for the experiment. So we have to be in Omega, always ( We love Omega ) P(Ω)=1 so yes, we are in Ω

Now, what the hell do I mean, by 'we are in' or 'we are not in' ? And this is important so focus! The experiment is a 2 coin toss experiment. We did the first toss and there will be another toss. We are talking about results here. So when I said, we are in H1, what I meant was that after the experiment completes, we are going to be in one of the elements of H1.

Okay so after the first toss ( which is heads as per our assumption ) if I ask. Are we in (HT,TH) ? What can we say ? Well, clearly, we can't reach TH but we can't say anything about HT that can only happen if we know what happened after the 2nd toss.

Cool Right ? No ? okay so how about to impress you further, I say not only if the first toss is heads, even if the first toss was tails, we could do that same thing with I1 and ( this is the killer ) if we take any 2 elements of I1 and Union them, or Intersect them, we will end up with an element of I1 again.

For example H1∪T1=Ω$H_1\cup T_1=\mathrm{\Omega }$ and H1∩Ω=H1$H_1\cap \mathrm{\Omega }=H_1$ and 

okay I think you get the point.

So now, what is the information after the 2nd well

I2=(ϕ,H1,HH,HT,T1,TH,TT,Ω)$$$$

think about it, you have tossed the coin tiwce and know both the tosses then you can tell if you are in or not in any of the sets of I2. Don't believe me ? Say you are get Heads in 1sttoss and Tails in the 2nd toss then you know for sure that you are, or are not in say, HH(spoiler: you aren't!)