Assembly of Measure Spaces: Restriction, Transport, One-Point Spaces, and Countable Disjoint Unions

Measurability of maps between measurable spaces is that of Measurable Function and Real-Valued Measurable Function; measurability and integrals of $[0,\infty]$-valued functions are those of Lebesgue Integral of a Nonnegative Measurable Function; measures use the conventions of Measure, Measure Space, and Probability Measure. A set is called countable here if it is finite or is the set of values of a sequence (which need not be injective). For a countable index set $I$ and members $a_i\in[0,\infty]$, the sum $\sum_{i\in I}a_i$ denotes the least upper bound of the finite partial sums $\sum_{i\in J}a_i$ over finite subsets $J\subseteq I$; when $I$ is the set of natural numbers, the terms being nonnegative makes the initial-segment partial sums cofinal among the finite partial sums, so this agrees with the sequential sum used in the countable additivity axiom of Measure, Measure Space, and Probability Measure. Let $(X,\mathcal{F},\mu)$ be a measure space.

1. (Restriction) Let $X_0\in\mathcal{F}$. Then $\mathcal{F}|_{X_0}=\{A\in\mathcal{F}:A\subseteq X_0\}$ is a $\sigma$-algebra on $X_0$, and setting $\mu|_{X_0}(A)=\mu(A)$ for $A\in\mathcal{F}|_{X_0}$ defines a measure, called the restriction of $(X,\mathcal{F},\mu)$ to $X_0$. A function $f:X_0\to[0,\infty]$ is measurable with respect to $\mathcal{F}|_{X_0}$ if and only if its zero extension $\tilde f$ (equal to $f$ on $X_0$ and to $0$ off $X_0$) is measurable with respect to $\mathcal{F}$, and in that case $\int_{X_0}f\,d\mu|_{X_0}=\int_X\tilde f\,d\mu.$

2. (Transport) Let $\varphi:X\to Z$ be a bijection onto a set $Z$. Then $\varphi(\mathcal{F})=\{\varphi(A):A\in\mathcal{F}\}$ is a $\sigma$-algebra on $Z$; the map $\varphi$ is measurable from $(X,\mathcal{F})$ to $(Z,\varphi(\mathcal{F}))$ and its inverse is measurable in the reverse direction; and the image measure $\mu_\varphi$ of $\mu$ under $\varphi$ satisfies $\mu_\varphi(\varphi(A))=\mu(A)$ for every $A\in\mathcal{F}$. The measure space $(Z,\varphi(\mathcal{F}),\mu_\varphi)$ is called the transport of $(X,\mathcal{F},\mu)$ along $\varphi$.

3. (One-point spaces) Let $Z=\{z\}$ be a one-point set. Then $\{\emptyset,Z\}$ is a $\sigma$-algebra on $Z$, and $\delta(\emptyset)=0$, $\delta(Z)=1$ defines a measure on it, called the one-point measure space with unit mass at $z$. Every function $g:Z\to[0,\infty]$ is measurable, and $\int_Z g\,d\delta=g(z)$.

4. (Countable disjoint union) Let $I$ be a nonempty countable set and, for each $i\in I$, let $(X_i,\mathcal{F}_i,\mu_i)$ be a measure space, the sets $X_i$ being pairwise disjoint. Put $X_\sqcup=\bigcup_{i\in I}X_i$, $\mathcal{F}_\sqcup=\{A\subseteq X_\sqcup:\ A\cap X_i\in\mathcal{F}_i\ \text{for every}\ i\in I\},\qquad \mu_\sqcup(A)=\sum_{i\in I}\mu_i(A\cap X_i)\quad(A\in\mathcal{F}_\sqcup),$ the sum in the finite-partial-sum sense above. The triple $(X_\sqcup,\mathcal{F}_\sqcup,\mu_\sqcup)$ is called the countable disjoint union of the family. Then:

(a) $\mathcal{F}_\sqcup$ is a $\sigma$-algebra on $X_\sqcup$; every member of every $\mathcal{F}_i$ belongs to $\mathcal{F}_\sqcup$; $\mu_\sqcup$ is a measure with $\mu_\sqcup(A)=\mu_i(A)$ whenever $A\in\mathcal{F}_i$; the restriction of the disjoint union to $X_i$ is $(X_i,\mathcal{F}_i,\mu_i)$; and if every $\mu_i(X_i)$ is finite, then $X_\sqcup$ is the union of countably many members of $\mathcal{F}_\sqcup$ of finite measure.

(b) A map $W:Y\to X_\sqcup$ from a measurable space $(Y,\mathcal{G})$ is measurable if and only if $W^{-1}(A)\in\mathcal{G}$ for every $i\in I$ and every $A\in\mathcal{F}_i$.

(c) A function $f:X_\sqcup\to[0,\infty]$ is measurable if and only if each restriction $f|_{X_i}$ is measurable with respect to $\mathcal{F}_i$, and in that case $\int_{X_\sqcup}f\,d\mu_\sqcup=\sum_{i\in I}\int_{X_i}f|_{X_i}\,d\mu_i.$