I'm taking a signal analysis course with a strong emphasis on probability. Long story short, the professor is assigning homework from the text which he said was optional, and expects us to learn the material straight out of that text. Many of these problems have to do with probability spaces, something which I feel completely lost with, not to mention set theory in general.

**I have here an example problem but would love if someone could help explain a solution in very general terms applying to probability spaces as a whole. Please bear in mind I'm basically floundering in the water until my textbook comes in the mail, and would appreciate a very simple explanation.**

**The Problem:**

A receiver listens to the data stream of zeros until the first one appears.

Then

$S = (0,01, 001, ...)$.

Choose

$A = 2^S$

and show that

$(S, A, P)$

is a probability **space**. Use your own probability measure $P$.

**My approach:**

I'm dimly aware of the criteria for a probability **space**, but I'm probably wrong somewhere:

$S$ must be a set:

**Check.**$A$ must be a sigma-algebra of $S$:

i) $S \in A$ :

**What****does**it**mean**if $S$ is in $2^S$? How is this expressed in terms of sets?ii) $ x \in A \Rightarrow \bar{x} \in A $:

**What would be the complement of some element in $A$ here? My best guess would be the next element in the sequence; i.e. the complement of 001 would be 0001 because the sequence continues if a 0 is chosen in the third place instead of a one. However, that also works for the infinite elements following 001, and surely they can't***all*be complements?iii) $ x, y \in A \Rightarrow x \cup y \in A$:

**I'm not sure how to express a union here. Isn't there only one set, the "union" of which would be itself?**$P$ must be a (countably additive) probability measure of $S$:

**The biggest issue here is not with the criteria but the actual construction of this probability measure. It seems to me that each term in the series has half the probability of the one preceding it; i.e. the $nth$ term has a $1/(2^n)$ chance of defining the actual event. This is a geometric series adding to 1, and links to the****definition**of the sigma-algebra from before, but how do I express this as a probability measure?i) $P(S) = 1$ :

**Check.**ii) $P($any element of $A) \geq 0 $:

**Check.**iii) $P(A \cup B) = P(A) + P(B), A \cap B = \varnothing$:

**What is the significance of A and B here? An example would be very helpful.**

**Thanks for any help you can offer, and sorry if the formatting is bad. I made an account and learned a bit of LaTex just to post this conundrum!**