Measure Spaces and the Lebesgue Integral: Standing Notation

This setting fixes the standing notation used by results about measure spaces and the Lebesgue integral. It is layered on The Real Numbers: Standing Notation and Background, whose notation for the real numbers, natural numbers, finite sums, sequences and bounds is in force throughout. It introduces no new concept and asserts nothing beyond the identifications recorded below, each of which is justified by the reference attached to it.

1. (The extended half-line)¶ $[0,\infty]$ denotes the set introduced in Measure, Measure Space, and Probability Measure, with the conventions for addition, multiplication and order fixed there, in particular $a+\infty=\infty+a=\infty$ and $0\cdot\infty=\infty\cdot 0=0$. A product in which one factor is $\infty$ and the other is positive is read as in Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$, which extends those conventions by $a\cdot\infty=\infty\cdot a=\infty$ for $0<a\le\infty$; the two prescriptions agree wherever both apply, since the convention of Measure, Measure Space, and Probability Measure covers only the factor $0$. The sum $\sum_{m\in\mathbb{N}}a_{m}$ of a sequence in $[0,\infty]$ is the one defined in Measure, Measure Space, and Probability Measure, and least upper bounds and greatest lower bounds of families in $[0,\infty]$ are read as in Monotone Convergence Theorem and Fatou's Lemma.

2. (Measure space)¶ $(X,\mathcal{F},\mu)$ denotes a measure space: $\mathcal{F}$ is a $\sigma$-algebra on the set $X$ and $\mu:\mathcal{F}\to[0,\infty]$ is a measure. The words finite, $\sigma$-finite and probability measure are used as defined there, and the elementary properties of Basic Properties of a Measure are in force.

3. (Measurable functions)¶ A map $f:X\to\mathbb{R}$ is measurable when it is measurable with respect to $\mathcal{F}$ and the Borel $\sigma$-algebra of the real line, which by claim 3 of Rational Intervals and Rays Generate the Borel Sigma-Algebra of the Real Line holds exactly when $\{x\in X:f(x)>c\}\in\mathcal{F}$ for every real $c$; a map $f:X\to[0,\infty]$ is measurable when it satisfies that same condition, as in Lebesgue Integral of a Nonnegative Measurable Function, the two readings agreeing for real-valued maps. The arithmetic of measurable real-valued maps is that of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions.

4. (Simple functions)¶ For $A\subseteq X$ the indicator $\mathbf{1}_{A}$, the notion of a simple function on $(X,\mathcal{F})$, its standard representation, and the integral of a nonnegative simple function are those of Simple Function and Its Integral.

5. (The integral)¶ For measurable $f:X\to[0,\infty]$, $\int_{X}f\,d\mu\in[0,\infty]$ is the integral of Lebesgue Integral of a Nonnegative Measurable Function. A measurable $f:X\to\mathbb{R}$ is integrable, with integral $\int_{X}f\,d\mu\in\mathbb{R}$, in the sense of Integrable Function and the Lebesgue Integral; by the criterion recorded there this holds exactly when $\int_{X}|f|\,d\mu<\infty$.

6. (Null sets and almost everywhere)¶ A subset of $X$ is null if it is $\mu$-null, and a property of points of $X$ holds almost everywhere, abbreviated a.e., if it holds $\mu$-almost everywhere; the measure is always the $\mu$ of clause 2.

7. (Powers)¶ $\mathbb{R}_{+}$ denotes the set of nonnegative real numbers, as in Properties of Real Powers of Nonnegative Real Numbers. For $t\in\mathbb{R}_{+}$ and a positive real $a$, $t^{a}$ is the power of $t$ with exponent $a$, whose properties are those of Properties of Real Powers of Nonnegative Real Numbers.

8. (Background)¶ The following results are in force by reference: Linearity and Monotonicity of the Lebesgue Integral, Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem, The Integral of an Indicator Function is the Measure of the Set and The Real Vector Space of Real-Valued Functions on a Set.