# Equivalence of Definitions of Sigma-Algebra

## 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: | \(\displaystyle X \in \Sigma \) | |||||

\((\text {SA} 2)\) | $:$ | Closure under Complement: | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \relcomp X A \in \Sigma \) | ||||

\((\text {SA} 3)\) | $:$ | Closure under Countable Unions: | \(\displaystyle \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\displaystyle \bigcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |

### 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: | \(\displaystyle X \in \Sigma \) | |||||

\((\text {SA} 2')\) | $:$ | Closure under Complement: | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \relcomp X A \in \Sigma \) | ||||

\((\text {SA} 3')\) | $:$ | Closure under Countable Disjoint Unions: | \(\displaystyle \forall A_n \in \Sigma: n = 1, 2, \ldots:\) | \(\displaystyle \bigsqcup_{n \mathop = 1}^\infty A_n \in \Sigma \) |

### 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 $\displaystyle \bigcup_{n \mathop \in \N} E_n$ is also measurable by $(\text {SA} 2)$ of definition $1$.

$\Box$

### 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: | \(\displaystyle X \in \Sigma \) | |||||

\((\text {AS} 2)\) | $:$ | Closure under Union: | \(\displaystyle \forall A, B \in \Sigma:\) | \(\displaystyle A \cup B \in \Sigma \) | ||||

\((\text {AS} 3)\) | $:$ | Closure under Complement Relative to $X$: | \(\displaystyle \forall A \in \Sigma:\) | \(\displaystyle \relcomp X A \in \Sigma \) |

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*: $\S 1.2$