Summability of truncated normalized variances

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.

Change of variables for expectation: Let $X:(\mathrm{\Omega },\mathcal{F},\mathbb{P})\to (S,\mathrm{\Sigma })$ be a random element, let ${\mathbb{P}}_X$ be its law, and let $g:(S,\mathrm{\Sigma })\to \mathbb{R}$ or $g:(S,\mathrm{\Sigma })\to \mathbb{C}$ be measurable.

1. If $g\ge 0$, then $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$
2. If $g(X)$ is integrable, then $g$ is integrable with respect to ${\mathbb{P}}_X$ and the same formula holds: $\mathbb{E}[g(X)]={\int }_{S}gd{\mathbb{P}}_X\mathrm{.}$

Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$: Identify $\mathbb{Z}$ with its canonical copy inside $\mathbb{R}$, along the embeddings $\mathbb{N}\to \mathbb{Z}\to \mathbb{Q}\to \mathbb{R}$ (lem-nat-embeds-int, lem-int-embeds-rat, lem-rat-embeds-dense, def-integers). Then for every real $x$ there is exactly one integer $m$ with

$m\le x<m+1.$

It is written $\lfloor x\rfloor$ and called the integer part, or floor, of $x$.

Two independent ingredients are needed and neither may be dropped. Existence is the Archimedean property (thm-of-archimedean) together with the well-ordering of $\mathbb{N}$ (thm-well-ordering-principle): the first says that $x$ is caught between two integers at all, the second picks the least integer above $x$. Uniqueness is the discreteness of $\mathbb{Z}$: no integer lies strictly between $m$ and $m+1$.

This lemma is stated once here and reused. It is what turns "the nearest integer to $x$" from a picture into an object, and the companion page's oscillator $\psi (x)={\inf }_{n\in \mathbb{Z}}\mathrm{\mid }x-n\mathrm{\mid }$ is computed from it in one line.

Monotone convergence for the integral: Let $0\le f_1\le f_2\le \cdots$ be measurable and suppose $f_n(x)\uparrow f(x)$ for every $x$. Then $\int f_{n}d\mu \uparrow \int fd\mu \mathrm{.}$