# Cardinality of Infinite Sigma-Algebra is at Least Cardinality of Continuum

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## Theorem

Let $\MM$ be an infinite $\sigma$-algebra on a set $X$.

Then $\MM$ is has cardinality at least that of the cardinality of the continuum $\mathfrak c$:

- $\map \Card \MM \ge \mathfrak c$

### Corollary

Let $\MM$ be an infinite $\sigma$-algebra on a set $X$.

Then $\MM$ is uncountable.

## Proof

We first show that $X$ is infinite.

By the definition of a $\sigma$-algebra, $\MM$ is a subset of $\powerset X$.

Were $X$ finite, by Cardinality of Power Set of Finite Set, the cardinality of $\MM$ would be at most $2^{\map \Card X}$

As $2^{\map \Card X}$ is finite if $X$ is finite, $X$ must be infinite by hypothesis that $\MM$ is infinite.

By the definition of $\sigma$-algebra, $X \in \MM$.

Also, by Sigma-Algebra Contains Empty Set, $\O \in \MM$.

Construct a countable collection of sets $\family {F_1, F_2, F_3, \ldots}_\N$ as follows:

\(\ds F_1\) | \(=\) | \(\ds \O\) | |||||||||||||

\(\ds F_2\) | \(=\) | \(\ds X\) | |||||||||||||

We can continue this construction using the Axiom of Choice: | |||||||||||||||

\(\ds F_3\) | \(=\) | \(\ds \text {any set in } \MM \setminus \set {\O, X}\) | |||||||||||||

\(\ds \) | \(\vdots\) | \(\ds \) | |||||||||||||

\(\ds F_n\) | \(=\) | \(\ds \text {any set in } \MM \setminus \set {F_1, F_2, \ldots, F_{n - 1} }\) | |||||||||||||

\(\ds \) | \(\vdots\) | \(\ds \) |

Consider an arbitrary $\ds S \in \bigcup_{k \mathop \in \N} F_k$

Then $S \in F_k$ for some $F_k$.

By the Well-Ordering Principle, there is a smallest such $k$.

Then for any $j < k$, $S \notin F_j$

Thus the sets in $\family {F_i}$ are disjoint.

Recall $\MM$ is infinite.

Then by Relative Difference between Infinite Set and Finite Set is Infinite, $\MM \setminus \set {F_1, F_2, \cdots , F_{k - 1} }$ is infinite.

Thus this process can continue indefinitely, choosing an arbitary set in $\MM$ that hasn't already been chosen for an earlier $F_k$.

By the definition of a $\sigma$-algebra, $\ds \bigsqcup_{i \mathop \in \N} F_i$ is measurable.

By the definition of an indexed family, $\family {F_i}_{i \mathop \in \N}$ corresponds to a mapping $\ds \iota: \N \hookrightarrow \bigsqcup_{i \mathop \in \N} F_i$

Such a mapping $\iota$ is injective because:

- each $F_i$ is disjoint from every other

- each $F_i$ contains a distinct $x \in X$

- $X$ has infinitely many elements.

Define:

- $\iota^*: \powerset \N \to \MM$:

- $\ds \map {\iota^*} N = \bigsqcup_{i \mathop \in N} F_i$

That is, for every $N \subseteq \N$, $\map {\iota^*} N$ corresponds to a way to select a countable union of the sets in $\family {F_i}$.

Because any distinct $F_i, F_j$ are disjoint, any two distinct ways to create a union $\ds S \mapsto \bigsqcup_{i \mathop \in S} F_i$ will result in a different union $\map {\iota^*} S$.

Thus $\iota^*$ is an injection into $\MM$.

Then the cardinality of $\MM$ is at least $\powerset \N$.

From Power Set of Natural Numbers is Cardinality of Continuum, $\R \sim \powerset \N$.

Thus $\MM$ is uncountable and:

- $\map \Card \MM \ge \mathfrak c$

$\blacksquare$

## Axiom of Choice

This theorem depends on the Axiom of Choice.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.

## Sources

- 1984: Gerald B. Folland:
*Real Analysis: Modern Techniques and their Applications*: Exercise $1.3$