Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation

(i) This setting extends Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices: its conventions and notation are in force.

(ii) $\infty$, $-\infty$, the extended real line $[-\infty,\infty]$, the set $[0,\infty]$, and $\le$, $<$, $\ge$, $>$ on $[-\infty,\infty]$ are as in The Extended Real Line, Its Order, and [0,∞] (i), The Extended Real Line, Its Order, and [0,∞] (ii) and The Extended Real Line, Its Order, and [0,∞] (iii); for a subset $A$ of $[-\infty,\infty]$, $\sup A$ and $\inf A$ are its supremum and infimum in $[-\infty,\infty]$, which exist by Order and Suprema in the Extended Real Line (iii). The sum $+$ and the product $\cdot$ on $[0,\infty]$ are those of Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii), and for a sequence $(a_{m})$ in $[0,\infty]$, $\sum_{m\in\mathbb{N}}a_{m}$ is its sum. Background: Order and Suprema in the Extended Real Line, Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation, Suprema and Finite Sums in [0,∞], Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence, Small Facts about [0,∞]: Zero, Finiteness and Sums of Zeros and Sums of Sequences in [0,∞] as Typed Terms and as a Map.

(iii) Countable sets are as in Countable Set (i). Background on countable sets: Countable Sets: a Bijection from ℕ onto ℕ×ℕ, Finite Sets, Subsets, Surjective Images, Images, Products, and the Integers and Rationals; on sequences of sets: Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset, Monotone Sequences of Sets, Unions of Traces, and Disjointness of Smaller Sets, The Disjointing Sequence of a Sequence of Sets; on induction and the order of $\mathbb{N}$: Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1.

(iv) For a set $X$, $\sigma$-algebras on $X$ and measurable spaces $(X,\mathcal{F})$ are as in σ-Algebra and Measurable Space; for a set $\mathcal{C}$ of subsets of $X$, $\sigma(\mathcal{C})$ is the $\sigma$-algebra on $X$ generated by $\mathcal{C}$; and $\pi$-systems and $\lambda$-systems on $X$ are as in π-System (i) and λ-System (i). Background: Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra, Dynkin's Pi-Lambda Theorem and Predicative Formulas for σ-Algebras, π-Systems and λ-Systems, and the Sets of Them Containing a Given Family.

(v) For a metric space $(X,d)$, $\mathcal{B}(X,d)$, or $\mathcal{B}(X)$, is its Borel $\sigma$-algebra, whose members are its Borel subsets; for $n,p,q\in\mathbb{N}$, $\mathcal{B}(\mathbb{R})$, $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathcal{M}_{p\times q}(\mathbb{R}))=\mathcal{B}(\mathbb{R}^{p\times q})$ are those of Borel Sigma-Algebra of a Metric Space (ii), Borel Sigma-Algebra of a Metric Space (iii) and Borel Sigma-Algebra of a Metric Space (iv). Background: Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces and Intervals Are Borel, and Rays, Half-Open Intervals and Rational Intervals Generate the Borel Sigma-Algebra of the Real Line.

(vi) For a measurable space $(X,\mathcal{F})$, a measure $\mu$ on $(X,\mathcal{F})$, the measure space $(X,\mathcal{F},\mu)$, and finite, $\sigma$-finite and probability measures are as in Measure, Measure Space, and Probability Measure. Background: Basic Properties of a Measure, Uniqueness of Measures Agreeing on a Generating π-System and Restricting a Measure to a Measurable Set.

(vii) For a measure space $(X,\mathcal{F},\mu)$, $\mu$-null subsets of $X$ and properties holding $\mu$-almost everywhere ($\mu$-a.e.) are as in Null Set of a Measure (i) and A Property Holding Almost Everywhere (i). Background: Null Sets and Properties Holding Almost Everywhere.

(viii) For a set $X$, outer measures $\mu^{*}$ on $X$ are as in Outer Measure (i); for such $\mu^{*}$, $\mu^{*}$-measurable subsets of $X$ are as in Carathéodory Measurable Sets of an Outer Measure (i), and $\mathcal{M}(\mu^{*})$ is the set of them of Carathéodory Measurable Sets of an Outer Measure (ii). Background: Carathéodory's Theorem: the Measurable Sets of an Outer Measure Form a σ-Algebra on Which It Is a Measure.

(ix) $\lambda^{*}:\mathcal{P}(\mathbb{R})\to[0,\infty]$ is the Lebesgue outer measure on $\mathbb{R}$ and $\lambda:\mathcal{B}(\mathbb{R})\to[0,\infty]$ is Lebesgue measure on $\mathbb{R}$. Background: Properties of Lebesgue Outer Measure on the Real Line and Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness. The axiom of dependent choice is not a standing hypothesis of this setting: Properties of Lebesgue Outer Measure on the Real Line (vi) and Properties of Lebesgue Outer Measure on the Real Line (vii), and Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (ii), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iii), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (iv), Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (v) and Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (vi), among them those by which $\lambda^{*}$ is an outer measure and $\lambda$ a measure, assume it, while Lebesgue Measure on the Real Line: Existence, Lengths of Intervals and Uniqueness (i) does not.