Interval realization from refining small diameter partitions

A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included: Let $n\ge 1$, assume the Axiom of Countable Choice (def-countable-choice), and let $a_i\le b_i$ be reals for $i<n$. Write

$R^{\circ }:=\{x\in {\mathbb{R}}^n:a_i<x_i<b_i\text{ for every }i<n\},\bar{R}:=[a,b]=\{x\in {\mathbb{R}}^n:a_i\le x_i\le b_i\text{ for every }i<n\}$

(def-multidimensional-rectangle-and-volume). Then $R^{\circ }$ is open and $\bar{R}$ is closed, so both are Borel and Lebesgue measurable, and every set $R$ with $R^{\circ }\subseteq R\subseteq \bar{R}$ is Lebesgue measurable with

${\lambda }_n(R)={\prod }_{i<n}(b_i-a_i)\mathrm{.}$

In particular this covers the four one-dimensional face conventions in each coordinate — the open box, the closed box $[a,b]$, the half-open box $B(a,b)={\prod }_{i<n}(a_i,b_i]$ of def-half-open-box, and every mixture of them, in any combination of coordinates — and it gives measure $0$ to all of them whenever $a_i=b_i$ for some $i<n$. For a half-open box with infinite parameters the value is already ${\lambda }_n(B)=\mathrm{vol}(B)$ (thm-lebesgue-measure-is-a-complete-measure).

Every at most countable subset of ${\mathbb{R}}^n$ is Lebesgue null; in particular ${\lambda }_1(\mathbb{Q})=0$: Let $n\ge 1$ and assume the Axiom of Countable Choice (def-countable-choice). Every at most countable subset $E\subseteq {\mathbb{R}}^n$ (def-countable) is Lebesgue measurable with

${\lambda }_n(E)=0,$

so $E$ is a ${\lambda }_n$-null set (def-measure-null-set-and-almost-everywhere). In particular every singleton is null, and on the real line the set ${\mathbb{Q}}_{\mathbb{R}}$ of rational reals (lem-rat-embeds-dense) satisfies ${\lambda }_1({\mathbb{Q}}_{\mathbb{R}})=0$.

Complete metric space: every Cauchy sequence converges in the space: Let $(X,d)$ be a metric space (def-metric-space).

$(X,d)$ is complete if every Cauchy sequence in $(X,d)$ (def-cauchy-in-metric) converges to a point of $X$ (def-metric-convergence).

A subset $A\subseteq X$ is called complete when the metric subspace $(A,d_A)$ is complete (def-isometry-and-metric-embedding); as always, the metric is part of the data, and $d_A$ is the restriction of $d$ to $A\times A$.

The limit is unique when it exists, since limits in a metric space are unique (lem-metric-limits-unique), so a complete space assigns to each of its Cauchy sequences one point and not a set of points.

Completeness is a property of the pair $(X,d)$, not of $X$ and not of the topology of $d$. Both quantifiers in the definition are about the metric: the Cauchy condition is stated with distances, and so is convergence. Two metrics on the same set can have the same open sets while exactly one of them is complete, which is the content of fs-completeness-is-a-topological-property and its witness. Read the word complete as an abbreviation for complete with respect to this metric, always.

Dominated convergence: Let $f$ and $(f_n)$ be measurable complex-valued functions such that $f_n\to f$ almost everywhere and $\mathrm{\mid }{f}_n\mathrm{\mid }\le g$ almost everywhere for a single nonnegative measurable function $g$ with $\int gd\mu <+\mathrm{\infty }$. Then $f\in L^1(\mu )$, $\int \mathrm{\mid }{f}_n-f\mathrm{\mid }d\mu ⟶0,$ and hence $\int f_{n}d\mu ⟶\int fd\mu \mathrm{.}$

Weak limits are unique: Bounded continuous real tests determine Borel probability measures on any metric space. In particular, weak limits are unique.