Summable untruncated variances are not necessary

First Borel-Cantelli lemma for events: Let $(A_n{)}_{n\in \mathbb{N}}$ be events in a probability space. If ${\sum }_{n=0}^{\mathrm{\infty }}\mathbb{P}(A_n)<+\mathrm{\infty },$ then $\mathbb{P}(A_n\mathrm{i}\mathrm{.}\mathrm{o}\mathrm{.})=0.$ No independence hypothesis is needed.

Coordinate random elements of a countable product are independent: Under the measure of thm-countable-product-of-probability-spaces, the coordinate maps $X_n(x)=x_n$ have laws ${\mu }_n$ and are independent.

Assuming countable and dependent choice, countable products of arbitrary probability spaces: Assume countable choice and dependent choice. For probability spaces $(E_n,{\mathcal{E}}_n,{\mu }_n{)}_{n\in \mathbb{N}}$ there is a unique probability measure $\mu$ on ${\mathcal{C}}_{\mathbb{N}}$ such that, for every finite $F$, its $F$-coordinate marginal is ${⨂}_{n\in F}{\mu }_n$.

The p-series for a real exponent p converges exactly when p is greater than one: For every real $p$, ${\sum }_{k\ge 1}\frac{1}{k^p}\text{ converges}⟺p>1.$

Almost-sure convergence of a random series: For real random variables $(X_n{)}_{n\ge 1}$, the series ${\sum }_{n\ge 1}{X}_n$ converges almost surely if its partial sums $S_n$ converge to a finite real limit on an event of probability one, as in def-almost-sure-convergence-of-random-variables. With $S_0=0$ from def-partial-sums-and-sample-means, its convergence event is $C={\bigcap }_{r\ge 1}{\bigcup }_{N\ge 1}{\bigcap }_{j\ge i\ge N}\{\mathrm{\mid }{S}_j-S_i\mathrm{\mid }<1\mathrm{/}r\}\mathrm{.}$ This is exactly the real Cauchy condition, with the indexing of thm-series-cauchy-criterion shifted by one. Measurable arithmetic makes every event in this countable expression measurable. For any fixed $m$, the union over $N$ may be restricted to $N\ge m$; then each difference uses only $X_{m+1},X_{m+2},\dots$. Thus $C$ is in the tail sigma-algebra, without assuming independence. Under independence, cor-almost-sure-convergence-of-an-independent-series-is-a-zero-one-event gives $\mathbb{P}(C)\in \{0,1\}$. Set $S={\lim }_n{S}_n$ on $C$ and $S=0$ off $C$. The functions ${\mathbf{1}}_C{S}_n$ converge everywhere to $S$, so thm-sequential-suprema-infima-limsup-liminf-and-pointwise-limits-are-measurable and thm-arithmetic-and-lattice-operations-preserve-measurability make $S$ measurable. For Borel sets $B_n$, the event $\{X_n\in B_n\text{ infinitely often}\}={\bigcap }_m{\bigcup }_{n\ge m}\{X_n\in B_n\}$ is likewise tail measurable. Changing finitely many summands adds an eventually constant finite difference to $S_n$; divided by deterministic $c_n>0$ tending to infinity that difference tends to zero, so the normalized limsup is unchanged. The sign of the unnormalized limsup need not be unchanged: the all-zero sequence has limsup zero, while changing its first term to $1$ makes the limsup of partial sums equal to $1$.