Equivalence of Definitions of Sigma-Algebra
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Theorem
The following definitions of the concept of Sigma-Algebra are equivalent:
Definition 1
Let $X$ be a set.
A $\sigma$-algebra $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties:
\((\text {SA} 1)\) | $:$ | Unit: | \(\ds X \in \Sigma \) | |||||
\((\text {SA} 2)\) | $:$ | Closure under Complement: | \(\ds \forall A \in \Sigma:\) | \(\ds \relcomp X A \in \Sigma \) | ||||
\((\text {SA} 3)\) | $:$ | Closure under Countable Unions: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Definition 2
Let $X$ be a set.
A $\sigma$-algebra $\Sigma$ over $X$ is a system of subsets of $X$ with the following properties:
\((\text {SA} 1')\) | $:$ | Unit: | \(\ds X \in \Sigma \) | |||||
\((\text {SA} 2')\) | $:$ | Closure under Complement: | \(\ds \forall A \in \Sigma:\) | \(\ds \relcomp X A \in \Sigma \) | ||||
\((\text {SA} 3')\) | $:$ | Closure under Countable Disjoint Unions: | \(\ds \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\ds \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |
Definition 3
A $\sigma$-algebra $\Sigma$ is a $\sigma$-ring with a unit.
Definition 4
Let $X$ be a set.
A $\sigma$-algebra $\Sigma$ over $X$ is an algebra of sets which is closed under countable unions.
Proof
Definition 1 implies Definition 3
Let $\Sigma$ be a system of sets on a set $X$ such that:
- $(1): \quad X \in \Sigma$
- $(2): \quad \forall A, B \in \Sigma: \relcomp X A \in \Sigma$
- $(3): \quad \ds \forall A_n \in \Sigma: n = 1, 2, \ldots: \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma$
Let $A, B \in \Sigma$.
From the definition:
- $\forall A \in \Sigma: A \subseteq X$.
Hence from Intersection with Subset is Subset:
- $\forall A \in \Sigma: A \cap X = A$
Hence $X$ is the unit of $\Sigma$.
So by definition 2 of $\sigma$-ring it follows that $\Sigma$ is a $\sigma$-ring with a unit.
Thus $\Sigma$ is a $\sigma$-algebra by definition 3.
$\Box$
Definition 3 implies Definition 1
Let $\Sigma$ be a $\sigma$-ring with a unit $X$.
By definition, $X \in \Sigma$.
From definition 2 of $\sigma$-ring, $\Sigma$ is:
- $(1) \quad$ closed under set difference.
- $(2) \quad$ closed under countable union
From Unit of System of Sets is Unique, we have that:
- $\forall A \in \Sigma: A \subseteq X$
from which we have that $X \setminus A = \relcomp X A$.
So $\Sigma$ is a $\sigma$-algebra by definition 1.
$\Box$
Definition 1 implies Definition 2
Follows directly from the definitions, as a disjoint union is a type of union.
$\Box$
Definition 2 implies Definition 1
Let $\Sigma$ be a system of sets on a set $X$ such that:
- $(1): \quad X \in \Sigma$
- $(2): \quad \forall A, B \in \Sigma: \relcomp X A \in \Sigma$
- $(3): \quad \ds \forall A_n \in \Sigma: n = 1, 2, \ldots: \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma$
Conditions $(1)$ and $(2)$ in definition 2 are identical to that of conditions $(1)$ and $(2)$ in definition 1.
Let $\family {E_n}_{n \mathop \in \N}$ be a countable indexed family of sets in $\Sigma$.
By Union of Indexed Family of Sets Equal to Union of Disjoint Sets:
- $\ds \bigsqcup_{n \mathop \in \N} F_n = \bigcup_{n \mathop \in \N} E_n$
for an appropriately constructed countable indexed family of disjoint sets in $\Sigma$.
By the hypotheses of definition $2$:
- $\ds \bigsqcup_{k \mathop \in \N}^\infty F_k$
is measurable.
Thus $\ds \bigcup_{n \mathop \in \N} E_n$ is also measurable by $(\text {SA} 2)$ of definition $1$.
$\Box$
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Definition 1 implies Definition 4
Immediate from the definition of algebra along with the added condition of closure under countable unions.
$\Box$
Definition 4 implies Definition 1
By definition $1$ of algebra of sets, an algebra has the properties:
\((\text {AS} 1)\) | $:$ | Unit: | \(\ds X \in \Sigma \) | |||||
\((\text {AS} 2)\) | $:$ | Closure under Union: | \(\ds \forall A, B \in \Sigma:\) | \(\ds A \cup B \in \Sigma \) | ||||
\((\text {AS} 3)\) | $:$ | Closure under Complement Relative to $X$: | \(\ds \forall A \in \Sigma:\) | \(\ds \relcomp X A \in \Sigma \) |
Replacing $(\text {AS} 2)$ with closure under countable unions immediately yields the first definition.
$\blacksquare$
Sources
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras: Theorem $1.1.1$
- 1984: Gerald B. Folland: Real Analysis: Modern Techniques and their Applications : $\S 1.2$