Measurable Functions and the Lebesgue Integral: Notation

(i) Base. This setting extends Measure Spaces, Borel Sets and Lebesgue Measure on the Real Line: Standing Notation: its conventions and notation are in force. Below, $(X,\mathcal{F},\mu)$ is a measure space and $n,p,q$ are natural numbers; the notation is introduced for all of them at once, and a result adopting this setting uses it for whichever it names.

(ii) Measurable functions. For a measurable space $(Y,\mathcal{G})$, $\mathcal{F}/\mathcal{G}$-measurable functions $f:X\to Y$ and the set $\mathrm{Meas}(X,\mathcal{F};Y,\mathcal{G})$ of them are as in Measurable Map, and the Set of Measurable Maps (i) and Measurable Map, and the Set of Measurable Maps (ii); $\mathcal{B}([0,\infty])$ is the Borel $\sigma$-algebra of $[0,\infty]$. A function into $\mathbb{R}$, $\mathbb{R}^{n}$, $\mathbb{R}^{p\times q}$ or $[0,\infty]$ is measurable as a function into the target named, as in Measurable Map, and the Set of Measurable Maps (iii), Measurable Map, and the Set of Measurable Maps (iv), Measurable Map, and the Set of Measurable Maps (v) and Measurable Map, and the Set of Measurable Maps (vi). 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 and The Set of Coordinate Preimages of Borel Sets in ℝⁿ and of Entry Preimages in the Real Matrices.

(iii) Indicators and simple functions. For $A\subseteq X$, $\mathbf{1}_{A}$ is the indicator function of $A$. Simple functions, the sets $\mathcal{S}(X,\mathcal{F})$ of simple and $\mathcal{S}^{+}=\mathcal{S}^{+}(X,\mathcal{F})$ of nonnegative simple functions, and the simple integral $I_{\mu}:\mathcal{S}^{+}\to[0,\infty]$ are as in Simple Functions, Nonnegative Simple Functions, and the Simple Integral. For $g:X\to[0,\infty]$ and $m\in\mathbb{N}$, $s_{m}(g)$ is the dyadic approximation of $g$ of order $m$, and $(s_{m}(g))_{m\in\mathbb{N}}$ its sequence of dyadic approximations. 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) Integrals of nonnegative functions. $\mathcal{M}^{+}=\mathcal{M}^{+}(X,\mathcal{F})$ is the set of measurable functions into $[0,\infty]$, and $\int_{X}f\,d\mu=\int f\,d\mu\in[0,\infty]$ is the integral of $f\in\mathcal{M}^{+}$. For a sequence $(a_{m})$ in $[-\infty,\infty]$, $\liminf_{m\to\infty}a_{m}$ and $\limsup_{m\to\infty}a_{m}$ are its limit inferior and limit superior, formed from its tail infima and tail suprema. Background: Basic Properties of the Integral of Nonnegative Measurable Maps, Monotone Convergence Theorem and Fatou's Lemma; for the formal bookkeeping: The Strict Level Sets of a Sequence of Nonnegative Measurable Maps above a Simple Function Form a Sequence of Measurable Sets.

(v) Positive part, negative part and absolute value. For $f:X\to\mathbb{R}$, $f^{+}$, $f^{-}$ and $|f|$ are its positive part, negative part and absolute value.

(vi) Integrable functions. Integrable functions $X\to\mathbb{R}$, the set $\mathcal{L}^{1}(\mu)=\mathcal{L}^{1}(X,\mathcal{F},\mu)$ of them, and the integral $\int_{X}f\,d\mu=\int f\,d\mu\in\mathbb{R}$ of $f\in\mathcal{L}^{1}(\mu)$ are as in Integrable Functions and Their Lebesgue Integral (i), Integrable Functions and Their Lebesgue Integral (ii) and Integrable Functions and Their Lebesgue Integral (iii). Where a function lies in both $\mathcal{M}^{+}$ and $\mathcal{L}^{1}(\mu)$ and it matters, the text names which integral is meant; likewise for the integrals over $A$ of (ix).

(vii) Null sets and almost everywhere. $\mu$-null sets and properties holding $\mu$-almost everywhere are as in Null Set of a Measure (i) and A Property Holding Almost Everywhere (i); a property holds for $\mu$-almost every $x\in X$, or for almost every $x$, if it holds $\mu$-almost everywhere. Background: Null Sets and Properties Holding Almost Everywhere.

(viii) Pointwise operations and pointwise limits. Where $f,g$ are given as functions $X\to[0,\infty]$ (for instance as elements of $\mathcal{M}^{+}$) and $c\in[0,\infty]$, $f+g$, $cf$ and $fg$ are those of Sums, Multiples and Products of Functions into [0,∞] (i), Sums, Multiples and Products of Functions into [0,∞] (ii) and Sums, Multiples and Products of Functions into [0,∞] (iii). Where $f,g$ are given as functions $X\to\mathbb{R}$ and $a,b\in\mathbb{R}$, $f+g$, $fg$, $af$, $af+bg$ and $f-g$ are formed with the pointwise operations of Pointwise Operations on Real-Valued Maps (i). For $f:X\to[0,\infty]$ and $A\subseteq X$, $f\mathbf{1}_{A}$ is as in Sums, Multiples and Products of Functions into [0,∞] (iv). For a set $Y$, a sequence $(f_{m})$ of functions $X\to Y$ and $x\in X$, $(f_{m}(x))_{m}$ is the sequence of Evaluating a Sequence of Maps at a Point Gives a Sequence, and Its Range (i); a sequence $(a_{m})$ of real numbers converges to $L\in\mathbb{R}$, written $a_{m}\to L$, as in Convergent Sequences of Real Numbers (i).

(ix) Integrals over measurable subsets. For $A\in\mathcal{F}$, $(A,\mathcal{F}_{A},\mu_{A})$ is the restriction of $(X,\mathcal{F},\mu)$ to $A$. For $f\in\mathcal{M}^{+}$, $\int_{A}f\,d\mu\in[0,\infty]$ is the integral of $f$ over $A$; a measurable $f:X\to\mathbb{R}$ is integrable on $A$, and then $\int_{A}f\,d\mu\in\mathbb{R}$ is its integral over $A$, as in Integrals of Functions over a Measurable Subset. Background: Restricting a Measure to a Measurable Set.

(x) Vector- and matrix-valued functions. $\mathbb{R}^{n}$ and the real matrices are as in Euclidean Space and Real Matrices: Standing Notation (ii) and Euclidean Space and Real Matrices: Standing Notation (iii), and $\mathbb{R}^{p\times q}$ denotes $\mathcal{M}_{p\times q}(\mathbb{R})$, as in Borel Sigma-Algebra of a Metric Space (iv). For $f:X\to\mathbb{R}^{n}$ and $F:X\to\mathbb{R}^{p\times q}$, $f_{k}$ and $F_{ij}$ are their component and entry functions. The sets $\mathcal{L}^{1}(\mu;\mathbb{R}^{n})$ and $\mathcal{L}^{1}(\mu;\mathbb{R}^{p\times q})$ of integrable functions into $\mathbb{R}^{n}$ and $\mathbb{R}^{p\times q}$, and the integrals $\int_{X}f\,d\mu\in\mathbb{R}^{n}$ and $\int_{X}F\,d\mu\in\mathbb{R}^{p\times q}$ of their elements, are as in Integrable Vector- and Matrix-Valued Functions and Their Integrals (i), Integrable Vector- and Matrix-Valued Functions and Their Integrals (ii), Integrable Vector- and Matrix-Valued Functions and Their Integrals (iii) and Integrable Vector- and Matrix-Valued Functions and Their Integrals (iv).