Proof of Properties of Simple Functions and of the Simple Integral

Finite sums in $[0,\infty]$ are formed with $+$, which is associative and commutative with neutral element $0$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (i), as in Sums and Products over a Finite Set and over an Interval (i), Sums and Products over a Finite Set and over an Interval (ii) and Sums and Products over a Finite Set and over an Interval (iii). Subsets, Cartesian products and images of finite sets are finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (iv), Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v) and Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vi). Real $a,b\ge0$ lie in $[0,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (ii), and their sum and product in $[0,\infty]$ are the real ones, by Addition and Multiplication on [0,∞] (i) and Addition and Multiplication on [0,∞] (ii); by the latter also $0\cdot a=a\cdot0=0$ for $a\in[0,\infty]$, and $\mu(\emptyset)=0$ by Measure, Measure Space, and Probability Measure (i). Real sums with vanishing terms are evaluated by Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (iv), $\mathbb{R}$ being a field and so a commutative ring (The Structure of ℝ as a Field and an Ordered Ring (i), Fields (i)). For $u\in\mathcal{S}$ and $c\in\mathbb{R}$, $u^{-1}(\{c\})\in\mathcal{F}$, as $\{c\}\in\mathcal{B}(\mathbb{R})$ by Borel Sets of a Metric Space: Open and Closed Sets, Points, Open Sets of Subspaces, and Subspaces (iii), for the metric space of Borel Sigma-Algebra of a Metric Space (ii), and $u$ is measurable as a map into $\mathbb{R}$, by Measurable Map, and the Set of Measurable Maps (ii), Measurable Map, and the Set of Measurable Maps (iii) and Measurable Map, and the Set of Measurable Maps (i). Maps and families on a set given by an expression are formed as in Maps and Relations Given by Formulas (i), and the pair-indexed map $(c,d)\mapsto A_{c,d}$ below as in Maps and Relations Given by Formulas (ii); the fibre maps $c\mapsto u^{-1}(\{c\})$ for $u\in\mathcal{S}$ as in 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 (i), intersections $(V,W)\mapsto V\cap W$ of members of $\mathcal{P}(X)$ as in Preimages, Unions and Intersections over an Index Set, Suprema and Infima, and Superlevel Sets as Typed Terms (iii), the projections of $C\times D$ as in Projection, Evaluation and Currying Maps (i), and constant maps, maps combining the values of two maps, and families of maps as in Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii) and Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv).

(P) Two simple functions. Let $s,t\in\mathcal{S}^{+}$, and let $C=s[X]$ and $D=t[X]$, finite by Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii), so that $C\times D$ is finite. For $(c,d)\in C\times D$ let $A_{c,d}=s^{-1}(\{c\})\cap t^{-1}(\{d\})$, which lies in $\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix). By Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (ii), $x\in A_{c,d}$ if and only if $x\in X$ and $(c,d)=(s(x),t(x))$. Hence the sets $A_{c,d}$ are pairwise disjoint; for $c\in C$ the union of $(A_{c,d})_{d\in D}$ is $s^{-1}(\{c\})$, since $t(x)\in D$ for every $x\in X$, and for $d\in D$ the union of $(A_{c,d})_{c\in C}$ is $t^{-1}(\{d\})$. By Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iv) and Suprema and Finite Sums in [0,∞] (vi), $c\cdot\mu(s^{-1}(\{c\}))=\sum_{d\in D}c\cdot\mu(A_{c,d})$ for $c\in C$ and $d\cdot\mu(t^{-1}(\{d\}))=\sum_{c\in C}d\cdot\mu(A_{c,d})$ for $d\in D$. So the map $c\mapsto\sum_{d\in D}c\cdot\mu(A_{c,d})$ on $C$, of which Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (iv) sums the values, has the same values as the summand $c\mapsto c\cdot\mu(s^{-1}(\{c\}))$ of the simple integral and therefore equals it (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)), and likewise on $D$ for $t$. Summing over $c\in C$, respectively over $d\in D$, Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv) and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (iv) give

$$I_{\mu}(s)=\sum_{(c,d)\in C\times D}c\cdot\mu(A_{c,d}),\qquad I_{\mu}(t)=\sum_{(c,d)\in C\times D}d\cdot\mu(A_{c,d}).$$

Proof of (i). Let $A\in\mathcal{F}$. By Indicator Function of a Subset (i), $\mathbf{1}_{A}$ is a map $X\to\mathbb{R}$ with values in $\{0\}\cup\{1\}$, which is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (iii) and Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v); so $\mathbf{1}_{A}(x)\ge0$ for every $x\in X$ and $\mathbf{1}_{A}[X]$ is finite. Let $B\in\mathcal{B}(\mathbb{R})$. By Preimages under a Map: Membership, the Preimage Map, Sequences of Preimages, and the Set of Sets with Preimage in a Given Set (ii), $\mathbf{1}_{A}^{-1}(B)$ is $X$ if $0,1\in B$, is $A$ if $1\in B$ and $0\notin B$, is $X\setminus A$ if $0\in B$ and $1\notin B$, and is $\emptyset$ if $0,1\notin B$; each lies in $\mathcal{F}$ by σ-Algebra and Measurable Space (i) and Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (vi). So $\mathbf{1}_{A}$ is measurable by Measurable Map, and the Set of Measurable Maps (i) and Measurable Map, and the Set of Measurable Maps (iii), and $\mathbf{1}_{A}\in\mathcal{S}^{+}$ by Measurable Map, and the Set of Measurable Maps (ii), Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii) and Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii). Let $C=\mathbf{1}_{A}[X]$; in $I_{\mu}(\mathbf{1}_{A})$ of Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv) the term at $c=0$ is $0$. If $A=\emptyset$, then $C\subseteq\{0\}$, and $I_{\mu}(\mathbf{1}_{A})=0=\mu(A)$ by Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii). If $A\neq\emptyset$, then $C\subseteq\{0,1\}$, $1\in C$ and $\mathbf{1}_{A}^{-1}(\{1\})=A$, so every term at $c\neq1$ is $0$, and by Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iii) and Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (ii), $I_{\mu}(\mathbf{1}_{A})=1\cdot\mu(A)=\mu(A)$.

Proof of (ii). Let $s,t\in\mathcal{S}$ and $c\in\mathbb{R}$. By Measurable Map, and the Set of Measurable Maps (ii), $s$ and $t$ are measurable maps into $\mathbb{R}$, so $s+t$, $st$ and $cs$, whose values are those of Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i) by Pointwise Operations on Real-Valued Maps (i), and $\max(s,t)$ and $\min(s,t)$, whose values $\max\{s(x),t(x)\}$ and $\min\{s(x),t(x)\}$ are those of Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i), are measurable by that clause. Let $C=s[X]$ and $D=t[X]$, finite sets. The maps $s+t$, $st$ and $cs$ have values $F(s(x),t(x))$ at $x\in X$ for the maps $F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ with values $a+b$, $ab$ and $ca$ at $(a,b)$, respectively, so their images are finite by 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 (ii). For $x\in X$, $\max\{s(x),t(x)\}$ and $\min\{s(x),t(x)\}$ are elements of $\{s(x),t(x)\}$ by The Maximum and Minimum of Two Elements of a Total Order (i), so the images of $\max(s,t)$ and $\min(s,t)$ lie in $C\cup D$, which is finite. By Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii) the five maps belong to $\mathcal{S}$. If moreover $s,t\in\mathcal{S}^{+}$ and $c\ge0$, then for $x\in X$ the numbers $s(x)+t(x)$, $s(x)t(x)$ and $cs(x)$ are $\ge0$ (for the sum, $0=0+0\le s(x)+t(x)$ by Rules of Arithmetic and Order in an Ordered Field (iv), $\mathbb{R}$ being an ordered field by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum (i); for the products, by Ordered Rings (i), $\mathbb{R}$ being an ordered ring by The Structure of ℝ as a Field and an Ordered Ring (ii)), $\max\{s(x),t(x)\}\ge s(x)\ge0$ by The Maximum and Minimum of Two Elements of a Total Order (iii), and $\min\{s(x),t(x)\}\ge0$ by The Maximum and Minimum of Two Elements of a Total Order (v); so the five maps belong to $\mathcal{S}^{+}$ by Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii).

Proof of (iii). Let $U$ be the union of the family $(A_{i})_{i\in I}$. (R) For $x\in X$: if $x\in A_{j}$ with $j\in I$, then $\mathbf{1}_{A_{j}}(x)=1$ and $\mathbf{1}_{A_{i}}(x)=0$ for $i\neq j$, as $A_{i}\cap A_{j}=\emptyset$, so $s(x)=c_{j}$ by Indicator Function of a Subset (i) and Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (iv); if $x\notin U$, every term vanishes, and $s(x)=0$ by Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (iv) with the empty subset and Sums and Products over a Finite Set and over an Interval (ii). Hence $s(x)\ge0$ for every $x\in X$, and $s[X]$ is contained in the union of $\{0\}$ and the image of the map $i\mapsto c_{i}$ on $I$, finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (vi); this union is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (iii) and Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice (v), so $s[X]$ is finite. Each map $c_{i}\mathbf{1}_{A_{i}}$ is measurable by (i) and Arithmetic of Measurable Real- and [0,∞]-Valued Maps (i). If $I=\emptyset$, $s$ is the constant map with value $0$, by Sums and Products over a Finite Set and over an Interval (ii), and is measurable by Measurable Maps: Generators, Composition, Continuous Maps, Restrictions, and Criteria for Real- and [0,∞]-Valued Maps (iii). Otherwise $n=\#I\in\mathbb{N}$ by The Iterated Operation over a Finite Set Does Not Depend on the Enumeration (i), and there is a bijection $\varphi$ from $[n]$ onto $I$ by The Number of Elements of a Finite Set (i); then $s(x)=\sum_{k=1}^{n}c_{\varphi(k)}\mathbf{1}_{A_{\varphi(k)}}(x)$ for $x\in X$ by Sums and Products over a Finite Set and over an Interval (i), and $k\mapsto c_{\varphi(k)}\mathbf{1}_{A_{\varphi(k)}}$ is a map from $[n]$ to $\mathrm{Meas}(X,\mathcal{F};\mathbb{R},\mathcal{B}(\mathbb{R}))$ (formed with the map $A\mapsto\mathbf{1}_{A}$ of 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 (v)) by Measurable Map, and the Set of Measurable Maps (ii), so $s$ is measurable by Arithmetic of Measurable Real- and [0,∞]-Valued Maps (ii). Thus $s\in\mathcal{S}^{+}$ by Simple Functions, Nonnegative Simple Functions, and the Simple Integral (ii) and Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iii).

Let $C=s[X]$, and for $c\in C$ and $i\in I$ let $B_{c,i}=s^{-1}(\{c\})\cap A_{i}$, a member of $\mathcal{F}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix). By (R), $B_{c,i}=A_{i}$ if $c=c_{i}$ and $B_{c,i}=\emptyset$ if $c\neq c_{i}$. For $c\in C$ the family $(B_{c,i})_{i\in I}$ is pairwise disjoint with union $s^{-1}(\{c\})\cap U$, which is $s^{-1}(\{c\})$ if $c\neq0$, since $s(x)=0$ for $x\notin U$ by (R). So $c\cdot\mu(s^{-1}(\{c\}))=c\cdot\mu(s^{-1}(\{c\})\cap U)$ for every $c\in C$, both sides being $0$ if $c=0$, and this is $\sum_{i\in I}c\cdot\mu(B_{c,i})$ by Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iv) and Suprema and Finite Sums in [0,∞] (vi); so the map $c\mapsto\sum_{i\in I}c\cdot\mu(B_{c,i})$ on $C$ equals the summand $c\mapsto c\cdot\mu(s^{-1}(\{c\}))$ of the simple integral, as their values agree (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)). By Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv) and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (iv),

$$I_{\mu}(s)=\sum_{c\in C}\sum_{i\in I}c\cdot\mu(B_{c,i})=\sum_{i\in I}\sum_{c\in C}c\cdot\mu(B_{c,i}).$$

Let $i\in I$. For $c\in C$ with $c\neq c_{i}$, $c\cdot\mu(B_{c,i})=c\cdot\mu(\emptyset)=0$. If $A_{i}\neq\emptyset$, then $c_{i}\in C$ by (R), and the inner sum is $c_{i}\cdot\mu(A_{i})$ by Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (iii). If $A_{i}=\emptyset$, every term of the inner sum is $0$, and the inner sum is $0=c_{i}\cdot\mu(A_{i})$ by Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii). So the inner-sum map $i\mapsto\sum_{c\in C}c\cdot\mu(B_{c,i})$ on $I$ has the same values as the given map $i\mapsto c_{i}\cdot\mu(A_{i})$ and equals it (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)). Hence $I_{\mu}(s)=\sum_{i\in I}c_{i}\cdot\mu(A_{i})$.

Proof of (iv). Let $s,t\in\mathcal{S}^{+}$, with $C$, $D$ and $A_{c,d}$ as in (P). For $x\in X$, $x\in A_{c,d}$ exactly for $(c,d)=(s(x),t(x))$, so by Pointwise Operations on Real-Valued Maps (i), Indicator Function of a Subset (i) and Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (iv), $(s+t)(x)=s(x)+t(x)=\sum_{(c,d)\in C\times D}(c+d)\mathbf{1}_{A_{c,d}}(x)$. As $c+d\ge0$ for $(c,d)\in C\times D$ and the sets $A_{c,d}$ are pairwise disjoint members of $\mathcal{F}$, (iii), applied with the finite set $C\times D$, gives $I_{\mu}(s+t)=\sum_{(c,d)\in C\times D}(c+d)\cdot\mu(A_{c,d})$. Here $(c+d)\cdot\mu(A_{c,d})=c\cdot\mu(A_{c,d})+d\cdot\mu(A_{c,d})$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (ii) and Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (iii), so the map $(c,d)\mapsto(c+d)\cdot\mu(A_{c,d})$ on $C\times D$ equals the map $(c,d)\mapsto c\cdot\mu(A_{c,d})+d\cdot\mu(A_{c,d})$ whose sum Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (v) evaluates, as their values agree (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv)); hence Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (v) and (P) give $I_{\mu}(s+t)=I_{\mu}(s)+I_{\mu}(t)$.

Proof of (v). Let $s\in\mathcal{S}^{+}$, $c\in\mathbb{R}$ with $c\ge0$, and $C=s[X]$. For $x\in X$, $x\in s^{-1}(\{d\})$ exactly for $d=s(x)$, so by Pointwise Operations on Real-Valued Maps (i), Indicator Function of a Subset (i) and Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (iv), $(cs)(x)=cs(x)=\sum_{d\in C}(cd)\mathbf{1}_{s^{-1}(\{d\})}(x)$. The sets $s^{-1}(\{d\})$, $d\in C$, are pairwise disjoint members of $\mathcal{F}$, and $cd\ge0$; as $c\in[0,\infty]$, (iii), Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (ii), Suprema and Finite Sums in [0,∞] (vi) and Simple Functions, Nonnegative Simple Functions, and the Simple Integral (iv) give

$$I_{\mu}(cs)=\sum_{d\in C}(cd)\cdot\mu(s^{-1}(\{d\}))=c\cdot\sum_{d\in C}d\cdot\mu(s^{-1}(\{d\}))=c\cdot I_{\mu}(s).$$

Proof of (vi). Let $s,t\in\mathcal{S}^{+}$ with $s\le t$, with $C$, $D$ and $A_{c,d}$ as in (P), and let $(c,d)\in C\times D$. If $A_{c,d}\neq\emptyset$, let $x\in A_{c,d}$; then $c=s(x)\le t(x)=d$, also in $[0,\infty]$ by The Extended Real Line, Its Order, and [0,∞] (iii), so $c\cdot\mu(A_{c,d})\le d\cdot\mu(A_{c,d})$ by Arithmetic in [0,∞]: Monoid Laws, Distributivity, Monotonicity and Cancellation (iv). If $A_{c,d}=\emptyset$, both sides are $0$. By (P) and Suprema and Finite Sums in [0,∞] (iv), $I_{\mu}(s)\le I_{\mu}(t)$.

Proof of (vii). Let $s\in\mathcal{S}^{+}$, $C=s[X]$, $E_{c}=s^{-1}(\{c\})\in\mathcal{F}$ for $c\in C$, and $\nu$ as stated. Let $A\in\mathcal{F}$. For $x\in X$, $x\in A\cap E_{c}$ holds for $c=s(x)$ if $x\in A$, and for no $c\in C$ if $x\notin A$; so by Pointwise Operations on Real-Valued Maps (i), Indicator Function of a Subset (i) and Sums over Finite Sets in a Commutative Ring and in an Ordered Field: Distributivity, Products of Sums, Vanishing Terms, Sums over Pairs, Expanding Products of Sums, Counting, Comparison, Monotonicity and the Triangle Inequality (iv), $(s\mathbf{1}_{A})(x)=s(x)\mathbf{1}_{A}(x)=\sum_{c\in C}c\,\mathbf{1}_{A\cap E_{c}}(x)$. The sets $A\cap E_{c}$, $c\in C$, are pairwise disjoint members of $\mathcal{F}$, by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix), so (iii) gives

$$\nu(A)=\sum_{c\in C}c\cdot\mu(A\cap E_{c})\qquad(A\in\mathcal{F}).$$

In particular, since $\emptyset\cap E_{c}=\emptyset$, $\nu(\emptyset)=\sum_{c\in C}c\cdot\mu(\emptyset)=0$ by Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (ii). Let $(A_{m})$ be a pairwise disjoint sequence in $\mathcal{F}$ with union $A$. For $c\in C$, $(A_{m}\cap E_{c})_{m\in\mathbb{N}}$ is a sequence (Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (ii)) in $\mathcal{F}$, by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (ix) and Sequences and Maps of Sets: Complements, Traces, Two-Term Sequences, the Disjointing Sequence, and Sequences in a Subset (v); it is pairwise disjoint with union $A\cap E_{c}$. By Measure, Measure Space, and Probability Measure (i) and Sums of Sequences in [0,∞]: Real Series, Linearity, Comparison, Rearrangement, Double Series and Monotone Convergence (ix), $c\cdot\mu(A\cap E_{c})=\sum_{m\in\mathbb{N}}c\cdot\mu(A_{m}\cap E_{c})$. With $b(c,m)=c\cdot\mu(A_{m}\cap E_{c})$ for $(c,m)\in C\times\mathbb{N}$, a map from $C\times\mathbb{N}$ to $[0,\infty]$, Finite Sums: Vanishing Terms, Finite Additivity of Measures, and Finite Sums of Series in [0,∞] (v) gives

$$\nu(A)=\sum_{c\in C}\sum_{m\in\mathbb{N}}b(c,m)=\sum_{m\in\mathbb{N}}\sum_{c\in C}b(c,m)=\sum_{m\in\mathbb{N}}\nu(A_{m}).$$

So $\nu$ is a measure on $(X,\mathcal{F})$ by Measure, Measure Space, and Probability Measure (i).