Confusion over Motivating Example for a $\sigma$ Algebra

I was attending a Probability theory lecture and the professor introduced to the Algebra of sets. His definition was the standard definition as seen in many texts and it read like this,

Let $$\Omega$$ be a sample space. Define an algebra $$\mathcal{A}$$ over $$\Omega$$ as the collection of subsets of $$\Omega$$ such that

1. $$\Omega \in \mathcal{A}$$

2. If $$A \in \mathcal{A} \implies A^c \in \mathcal{A}$$

3. If $$A_i \in \mathcal{A}$$ for $$i=1,2,\dots,n$$ , then $$\bigcup\limits_{i=1}^n A_i \in \mathcal{A}$$

He then said the algebra even though it captures or gives a structure for the study of random experiments most times there will be events which is of interest and won't fit into an Algebra and then gives an examples as follows.

Consider the infinite coin toss experiment until we see the first head. For which $$\Omega = \{H,TH,TTH,TTTTH, \dots \}$$. Now suppose we are interested in the "event" that the first head occurs in an even toss, i.e, we look at the event $$A = \{TH,TTTH,TTTTTH, \dots \}$$. Then he said such a kind of "event cannot be captured under an algebra structure- because we just have finite unions and intersections"

I don't understand clearly why this example motivates us to see an $$\sigma$$-algebra, allowing closure under countably infinite unions.

Any help would be deeply appreciated.

2 Answers

Let $$A_n$$ be the event that the first head occurs at toss $$2n$$ and let $$\mathcal A$$ be the algebra generated by them.

Then: $$A=\bigcup_{n=1}^{\infty}A_n$$

This is an "interesting event" for us but unfortunately $$A\notin\mathcal A$$.

We can repair this by working instead with the $$\sigma$$-algebra generated by the $$A_n$$.

This is a common example. I think you've overlooked that the set A has to be constructed from the following specific class of events: $$A_i$$ is the event "head in the i-th toss". This leads to the the following representation for A:

$$a \in A \Rightarrow \exists k \in \mathbb{N}: a =\overset{2k-1}{\underset{n=1}\cap} (A_n)^c \cap A_{2k}$$, as this is the event: an odd number (2k-1) of straight tails, followed by a head.

Which means that $$A = \overset{\infty}{\underset{k=1}\cup}(\overset{2k-1}{\underset{n=1}\cap} (A_n)^c\cap A_{2k})$$, which is clearly not representable with finite unions and intersections.