Measurable Functions and the Lebesgue Integral: Standing Notation

(i) Base. This setting extends Measurable Functions and the Lebesgue Integral: Notation: its conventions and notation, and through it those of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation, are in force.

(ii) Measurable functions. Notation as in Measurable Functions and the Lebesgue Integral: Notation (ii). Background: Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps, Arithmetic of Measurable Real- and [0,∞]-Valued Maps, Suprema, Infima, Limits Inferior and Limits Superior of Measurable [0,∞]-Valued Maps, Suprema, Pointwise Limits and the Convergence Set of Measurable Real-Valued Maps, Vector- and Matrix-Valued Measurable Maps: Components, Entries and Continuous Images, Sums, Products, Transposes, Dot Products and Norms of Measurable Vector- and Matrix-Valued Maps, Borel Subsets of [0,∞]: Rays, the Point ∞, and Traces on ℝ, The Borel Sets of ℝⁿ and of the Real Matrices Are Generated by the Coordinates and Preimages of Unions, Intersections and Complements; for the formal bookkeeping: Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set, Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms, Unions and Intersections over a Countable Index Term of Terms with Values in a σ-Algebra Are Again Such Terms, The Set of Coordinate Preimages of Borel Sets in ℝⁿ and of Entry Preimages in the Real Matrices and Evaluating a Sequence of Maps at a Point Gives a Sequence, and Its Range.

(iii) Indicators and simple functions. Notation as in Measurable Functions and the Lebesgue Integral: Notation (iii). Background: Properties of Simple Functions and of the Simple Integral and Approximation of Measurable Functions by Dyadic Simple Functions; for the formal bookkeeping: Operations over the Finite Images of Maps, and the Summands of the Simple Integral, Maps Used to Build the Integral: Fibres of a Map, Pointwise Combinations with Finite Image, Tail Extrema of a Sequence of Maps, the Integer Part, and Indicators and The Integers Lying in a Bounded Real Interval Form a Finite Set.

(iv) The integral. Notation as in Measurable Functions and the Lebesgue Integral: Notation (iv), Measurable Functions and the Lebesgue Integral: Notation (v), Measurable Functions and the Lebesgue Integral: Notation (vi), Measurable Functions and the Lebesgue Integral: Notation (vii) and Measurable Functions and the Lebesgue Integral: Notation (viii). Background: Basic Properties of the Integral of Nonnegative Measurable Maps, Monotone Convergence Theorem, Fatou's Lemma, Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, Dominated Convergence Theorem, Null Sets and Properties Holding Almost Everywhere and The Lebesgue Integral and Null Sets; for the formal bookkeeping: The Functions f ↦ f⁺ and f ↦ f⁻ on the Real-Valued Functions on a Set, and a Predicative Form of g = f⁺ and g = f⁻ and The Strict Level Sets of a Sequence of Nonnegative Measurable Maps above a Simple Function Form a Sequence of Measurable Sets.

(v) Integrals over measurable subsets. Notation as in Measurable Functions and the Lebesgue Integral: Notation (ix). Background: Integrals over Measurable Subsets and Restricting a Measure to a Measurable Set.

(vi) Vector- and matrix-valued functions. Notation as in Measurable Functions and the Lebesgue Integral: Notation (x). Background: Rules for Integrals of Vector- and Matrix-Valued Functions; for the formal bookkeeping: Sets and Functions Defined Componentwise on Families of Vector- and Matrix-Valued Functions.

(vii) Lebesgue measure on a closed interval. For $a,b\in\mathbb{R}$ with $a<b$ and the closed interval $[a,b]$, $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ is the restriction of $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$ to $[a,b]$, as in Measurable Functions and the Lebesgue Integral: Notation (ix), and $\mathcal{L}^{1}(\lambda_{[a,b]})$ and $\int f\,d\lambda_{[a,b]}$ are as named in Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval. Background: Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval, Continuity on a Subset Is Continuity on the Metric Subspace and A Finite Sum of Nonnegative Real Numbers in [0,∞] Is Their Real Sum. The axiom of dependent choice is not a standing hypothesis: Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval assumes it, as do the results of Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation (ix) by which $\lambda$ is a measure, so $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ is a measure space only under it.