Motivational example for introducing singular distributions fom Rosenthal

Question

I am studying Rosenthal´s book, and I am stuck on one example, where author is showing why it is neccesary to consider distributions diffrent from absolutely continuous or discrete.

Well, what I don´t understand is the fact, that $P(Y \in S)=1$ (by law of large numbers, I dont see how the average of partial sums converges to 2/3) and also why the measure of introduced set is $\lambda = 0.$

Here is the screen of the motivational example.

Thanks in advance for any help.

Answer 1

Since the $Z_n$ are i.i.d, by the strong-law of large numbers $$ \frac{d_1(Y)+\dotsb d_n(Y)}{n}=\frac{Z_1+\dotsb+Z_n}{n}\to EZ_1=P(Z_1=1)=\frac{2}{3} $$ almost surely as $n\to \infty$. Hence $P(Y\in S)=1.$