I have a question about the formal treatment of infinite product measures in probability calculus. Take for example the model of an infinite coin toss. If $(\Omega_i,\mathcal A_i,P_i)$ is the probability space of a single coin toss (that is $\Omega_i = \{0,1\}$), the sample space of an infinite coin toss is given by the infinite cartesian product
$\Omega = \Omega_1\times\Omega_1\times\ldots = \{0,1\}^\mathbb N$.
Now, my textbook says that the $\sigma$-algebra of that product space is given by the smallest $\sigma$-algebra that contains all cylinder sets, i.e. all sets of the form
$A = B_1 \times B_2 \times B_3 \ldots$
where $B_i \in \mathcal A_i$ for a finite number of $i$'s and $B_i = \Omega_i$ otherwise.
Now my question is: Does this $\sigma$-algebra also contain events where an infinite number of outcomes are explicitly specified, for example the event where ALL coins shows "Head" ($\omega = (1,1,...)$ with infinite many 1's). Or does it only contain events where only a finite number of outcomes is specified?
Let $a$ be some sequence and let $A_n=\prod_{j=1}^n \{a_j\}\times \prod_{k=n+1}^{\infty} \{0,1\}.$ Then, clearly, each $A_n$ is a cylinder. Furthermore, $\cap_{n=1}^{\infty} A_n=\{a\},$ so yes, the singletons are measurable.
Naturally, this generalises to the statement that $\prod_{j=1}^{\infty}A_j$ is a measurable element of the infinite product if each $A_j$ is measurable (of an arbitrary infinite product, not just the coin toss space).