Bounded centered convergent series have summable variances

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 $1$.

Disjoint groups of an independent sigma-algebra family remain independent: Let $({\mathcal{F}}_i{)}_{i\in I}$ be an independent family of sigma-algebras on a probability space, and let $J_0,\dots ,J_{m-1}\subseteq I$ be pairwise disjoint index sets. For each $r<m$, define ${\mathcal{G}}_r:=\sigma \left({\bigcup }_{i\in J_r}{\mathcal{F}}_i\right)\mathrm{.}$ Then the sigma-algebras ${\mathcal{G}}_0,\dots ,{\mathcal{G}}_{m-1}$ are independent.

For $n\ge 1$, if $X_0,\dots ,X_{n-1}$ are independent real random variables and $g_i:\mathbb{R}\to \mathbb{R}$ are Borel measurable with each $g_i(X_i)$ integrable, then ${\prod }_{i<n}{g}_i(X_i)$ is integrable and $\mathbb{E}[{\prod }_{i<n}{g}_i(X_i)]={\prod }_{i<n}\mathbb{E}[g_i(X_i)]$ in $\mathbb{R}$ (Expectations factor over finite products of independent random variables).

Variance and covariance identities for random variables: Let $X,Y$ be square-integrable real random variables on one probability space. Then $\mathrm{Var}(X)=\mathbb{E}[X^2]-\mathbb{E}[X{]}^2,$ $\mathrm{Cov}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]\mathrm{.}$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.

Continuity from below for measures: Let $(E_n{)}_{n\in \mathbb{N}}$ be an increasing sequence of measurable sets for a measure $\mu$, so $E_n\subseteq E_{n+1}$. Then $\mu \left({\bigcup }_{n\in \mathbb{N}}{E}_n\right)={\sup }_{n\in \mathbb{N}}\mu (E_n)\mathrm{.}$ No finiteness hypothesis is required.

Linearity, monotonicity, and the modulus bound for expectation: Let $X,Y$ be integrable real or complex random variables on one probability space. 1. For scalars $a,b$, $\mathbb{E}[aX+bY]=a\mathbb{E}[X]+b\mathbb{E}[Y]\mathrm{.}$ 2. If $X$ and $Y$ are real-valued and $X\le Y$ almost surely, then $\mathbb{E}[X]\le \mathbb{E}[Y]\mathrm{.}$ 3. $\mathrm{\mid }\mathbb{E}[X]\mathrm{\mid }\le \mathbb{E}[\mathrm{\mid }X\mathrm{\mid }]\mathrm{.}$