Uniqueness of Measures

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Theorem

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\GG \subseteq \powerset X$ be a generator for $\Sigma$; that is, $\Sigma = \map \sigma \GG$.

Suppose that $\GG$ satisfies the following conditions:

$(1):\quad \forall G, H \in \GG: G \cap H \in \GG$
$(2):\quad$ There exists an exhausting sequence $\sequence {G_n}_{n \mathop \in \N} \uparrow X$ in $\GG$


Let $\mu, \nu$ be measures on $\struct {X, \Sigma}$, and suppose that:

$(3):\quad \forall G \in \GG: \map \mu G = \map \nu G$
$(4):\quad \forall n \in \N: \map \mu {G_n}$ is finite


Then:

$\mu = \nu$


Alternatively, by Countable Cover induces Exhausting Sequence, the exhausting sequence in $(2)$ may be replaced by a countable $\GG$-cover $\sequence {G_n}_{n \mathop \in \N}$, still subject to $(4)$.


Proof 1

Define, for all $n \in \N$, $\DD_n$ by:

$\DD_n := \set {E \in \Sigma: \map \mu {G_n \cap E} = \map \nu {G_n \cap E} }$

Let us show that $\DD_n$ is a Dynkin system.


By Intersection with Subset is Subset, $G_n \cap X = G_n$, whence $(3)$ implies that $X \in \DD_n$.

Now, let $D \in \DD_n$. Then:

\(\ds \map \mu {G_n \cap \paren {X \setminus D} }\) \(=\) \(\ds \map \mu {G_n \setminus D}\) Intersection with Set Difference is Set Difference with Intersection
\(\ds \) \(=\) \(\ds \map \mu {G_n} - \map \mu {G_n \cap D}\) $\map \mu {G_n} < +\infty$, Set Difference and Intersection form Partition, Measure is Finitely Additive Function
\(\ds \) \(=\) \(\ds \map \nu {G_n} - \map \nu {G_n \cap D}\) $(3)$, $D \in \DD_n$
\(\ds \) \(=\) \(\ds \map \nu {G_n \cap \paren {X \setminus D} }\) Above reasoning in opposite order

Therefore, $X \setminus D \in \DD_n$.


Finally, let $\sequence {D_m}_{m \mathop \in \N}$ be a sequence of pairwise disjoint sets in $\DD_n$.

Then:

\(\ds \map \mu {G_n \cap \paren {\bigcup_{m \mathop \in \N} D_m} }\) \(=\) \(\ds \map \mu {\bigcup_{m \mathop \in \N} \paren {G_n \cap D_m} }\) Intersection Distributes over Union
\(\ds \) \(=\) \(\ds \sum_{m \mathop \in \N} \map \mu {G_n \cap D_m}\) $\mu$ is a measure
\(\ds \) \(=\) \(\ds \sum_{m \mathop \in \N} \map \nu {G_n \cap D_m}\) $D_m \in \DD_n$
\(\ds \) \(=\) \(\ds \map \nu {G_n \cap \paren {\bigcup_{m \mathop \in \N} D_m} }\) Above reasoning in opposite order

Therefore:

$\ds \bigcup_{m \mathop \in \N} D_m \in \DD_n$.

Thus, we have shown that $\DD_n$ is a Dynkin system.


Combining $(1)$ and $(3)$, it follows that:

$\forall n \in \N: \GG \subseteq \DD_n$

From $(1)$ and Dynkin System with Generator Closed under Intersection is Sigma-Algebra:

$\map \delta \GG = \map \sigma \GG = \Sigma$

where $\delta$ denotes generated Dynkin system.


By definition of $\map \delta \GG$, this means:

$\forall n \in \N: \map \delta \GG \subseteq \DD_n$

That is, for all $n \in \N$, $\Sigma \subseteq \DD_n \subseteq \Sigma$.

By definition of set equality:

$\Sigma = \DD_n$ for all $n \in \N$


Thus, for all $n \in \N$ and $E \in \Sigma$:

$\map \mu {G_n \cap E} = \map \nu {G_n \cap E}$


Now, from Set Intersection Preserves Subsets, $E_n := G_n \cap E$ defines an increasing sequence of sets with limit:

$\ds \bigcup_{n \mathop \in \N} \paren {G_n \cap E} = \paren {\bigcup_{n \mathop \in \N} G_n} \cap E = X \cap E = E$

from Intersection Distributes over Union and Intersection with Subset is Subset.

Thus, for all $E \in \Sigma$:

\(\ds \map \mu E\) \(=\) \(\ds \lim_{n \mathop \to \infty} \map \mu {G_n \cap E}\) Characterization of Measures: $(3)$
\(\ds \) \(=\) \(\ds \lim_{n \to \infty} \map \nu {G_n \cap E}\) $\forall n \mathop \in \N: E \in \DD_n$
\(\ds \) \(=\) \(\ds \map \nu E\) Characterization of Measures: $(3)$

That is to say, $\mu = \nu$.

$\blacksquare$


Proof 2

Define the set:

$\AA = \set {T \in \Sigma: \forall G \in \GG: \map \mu {G \cap T} = \map \nu {G \cap T} }$

By Intersection with Subset is Subset, it follows that $G \cap X = G$ (for all $G \in \GG$); therefore, $X \in \AA$ by assumption $(3)$.

Now, define the set:

$\Sigma' = \set {S \in \Sigma: \forall T \in \AA: \map \mu {S \cap T} = \map \nu {S \cap T} }$

It follows directly from the definitions that $\GG \subseteq \Sigma' \subseteq \Sigma$.

Note that because $X \in \AA$ (from before), it follows that $\mu {\restriction_{\Sigma'} } = \nu {\restriction_{\Sigma'} }$ (where $\restriction$ denotes restriction).

By the definition of the $\sigma$-algebra generated by $\GG$ and the equality of sets, it suffices to show that $\Sigma'$ is a $\sigma$-algebra over $X$.

This would then imply that $\Sigma = \Sigma'$, the desired result.


  • Proposition $1$. If $T \in \AA$ and $G \in \GG$, then $G \cap T \in \AA$.

Proof. Let $H \in \GG$ be arbitrary.

By the associativity of intersection, it follows that $H \cap \paren {G \cap T} = \paren {H \cap G} \cap T$.

The result follows because $H \cap G \in \GG$ by assumption $(1)$.

$\Box$


  • Proposition $2$. If $T \in \AA$ and $S \in \Sigma'$, then $S \cap T \in \AA$.

Proof. Let $G \in \GG$ be arbitrary.

By the associativity and commutativity of intersection, $G \cap \paren {S \cap T} = S \cap \paren {G \cap T}$.

The result follows by the definition of $\Sigma'$, and because $G \cap T \in \AA$ by Proposition $1$.

$\Box$


  • Lemma $1$. If $A, B \in \Sigma'$, then $A \cap B \in \Sigma'$.

Proof. Let $T \in \AA$ be arbitrary.

By the associativity of intersection, it follows that $\paren {A \cap B} \cap T = A \cap \paren {B \cap T}$.

The result follows because $B \cap T \in \AA$ by Proposition $2$.

$\Box$


  • Proposition $3$. If $A, B \in \Sigma'$ and $\map \mu A$ is finite, then $A \cup B \in \Sigma'$.

Proof. Let $T \in \AA$ be arbitrary. Then:

\(\ds \map \mu {\paren {A \cup B} \cap T}\) \(=\) \(\ds \map \mu {\paren {A \cap T} \cup \paren {B \cap T} }\) Intersection is Commutative and Intersection Distributes over Union
\(\ds \) \(=\) \(\ds \map \mu {A \cap T} + \map \mu {B \cap T} - \map \mu {\paren {A \cap T} \cap \paren {B \cap T} }\) Measure is Strongly Additive
\(\ds \) \(=\) \(\ds \map \mu {A \cap T} + \map \mu {B \cap T} - \map \mu {\paren {A \cap B} \cap T}\) Intersection is Associative, Intersection is Commutative and Intersection is Idempotent

and similarly for $\nu$.

The result follows because $A \cap B \in \Sigma'$ by Lemma $1$.

$\Box$


  • Corollary $1$.
$\displaystyle \forall n \in \N: \bigcup_{k \mathop = 0}^n G_k \in \Sigma'$

Proof. The result follows by assumption $(4)$, Proposition $3$, the associativity of intersection, and mathematical induction on $n$.

$\Box$


  • Lemma $2$. If $S \in \Sigma'$, then $X \setminus S \in \Sigma'$.

Proof. Let $T \in \AA$ be arbitrary. Then:

\(\ds \map \mu {\paren {X \setminus S} \cap T}\) \(=\) \(\ds \lim_{n \mathop \to \infty} \map \mu {\bigcup_{k \mathop = 0}^n G_k \cap T \setminus S}\) by assumption $(2)$, Intersection with Set Difference is Set Difference with Intersection, Intersection is Commutative, Intersection Distributes over Union, Set Difference is Right Distributive over Union, and by Characterization of Measures: $(3)$
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \paren {\map \mu {\bigcup_{k \mathop = 0}^n G_k \cap T} - \map \mu {\bigcup_{k \mathop = 0}^n G_k \cap S \cap T} }\) Set Difference and Intersection form Partition, Measure is Finitely Additive Function, Intersection is Associative, Intersection is Commutative; this expression is defined because of Intersection is Subset, the monotonicity of $\mu$, and the subadditivity of $\mu$, by assumption $(4)$

and similarly for $\nu$.

The result follows by Corollary $1$ and Lemma $1$.

$\Box$


Proof. The result follows from Lemmas $1$ and $2$, and De Morgan's laws.

$\Box$


Proof. By Corollary $2$, it suffices to show that $\Sigma'$ is a $\sigma$-ring.

Let $\sequence {S_k}_{k \mathop = 0}^\infty$ be a sequence of sets in $\Sigma'$, and let $T \in \AA$ be arbitrary.

By the Intersection Distributes over Union and Characterization of Measures: $(3)$:

$\ds \map \mu {\bigcup_{k \mathop = 0}^\infty S_k \cap T} = \lim_{n \mathop \to \infty} \map \mu {\bigcup_{k \mathop = 0}^n S_k \cap T}$

and similarly for $\nu$.

By Corollary $2$, it follows by the associativity of intersection and by mathematical induction on $n$ that:

$\ds \forall n \in \N: \bigcup_{k \mathop = 0}^n S_k \in \Sigma'$

Therefore, $\Sigma'$ is a $\sigma$-ring, and the theorem is proven.

$\blacksquare$


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