If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

Question

I am going over a tutorial in my real analysis course. There is an proof in which I don't understand some parts of it.

The proof relates to the following proposition:

($S$ - infinite $\sigma$-algebra on $X$) $\implies $ $S$ is uncountable.

Proof:

Assume: $S=\{A_{i}\}_{i=1}^{+\infty}$. $\forall x\in X: B_{x}:=\cap_{x\in A_{i}}A_{i}$. [Note: $B_{x}\in S$ $\impliedby$ ($B_{x}$ - countable intersection].

Lemma: $B_{x}\cap B_{y}\neq\emptyset \implies B_{x}=B_{y}$.

Proof(of lemma):

$z\in B_{x}\cap B_{y} \implies B_{z}\subseteq B_{x}\cap B_{y}$.

1.$x\not\in B_{z} \implies x\in B_{x}\setminus B_{z} \wedge B_{x}\setminus B_{z} \subset S \wedge B_{x}\setminus B_{z} \subset B_{x}$ (contradiction:$\space$ definition of $B_{x}$) $\implies$ $B_{z}=B_{x}$

2.$y\not\in B_{z} \implies y\in B_{y} \setminus B_{z} \space \wedge \space B_{y}\setminus B_{z} \subset S \space\wedge\space B_{y} \setminus B_{z}\subset B_{y} $(contradiction: definition of $B_{y}$) $\implies$ $B_{z}=B_{y}$ $\implies B_{x}=B_{y} \space \square$

Consider: $\{B_{x}\}_{x\in X}$. If: there are finite sets of the form $B_{x}$ then: $S$ is a union of a finite number of disjoint sets $\implies$ $S$ is finite $\implies$ there is an infinite number of sets of the form $B_{x}$. $\implies$ $|\bigcup\limits_{i\in A \subseteq\mathbb{N}}B_{x_{i}}| \geq \aleph_{0}$.(contradiction) $\square$

There are couple of things I don't understand in this proof:

1. Why the fact that we found a set ($B_{x}\setminus B_{z}$) in $S$ containing $x$ and is strictly contained in $B_{x}$ a contradiction ?

2. Why if there are only a finite number of different sets of the form $B_{x}$ then $S$ is a union of a finite number of disjoint sets and is finite ?

Answer 1

1. Because $B_x$ is supposed to be the intersection of all measurable sets containing $x$, but you've found a measurable set containing $x$ strictly inside $B_x$.

2. Because for any measurable set $T$, we have $T=\bigcup_{x\in T}B_x$. Thus, if there are $n$ distinct sets of the form $B_x$, then there are at most $2^n$ elements of $S$.

Answer 2

One can actually change the argument of @Josef slightly to obtain a correct proof. This is probably also what @ncmathsadist is hinting at, although he does not indicate how to obtain the countably infinite collection of disjoint subsets.

Incidentially, the argument below even shows that an infinite $\sigma$-algebra is not only uncountable, but it has at least the cardinality of the continuum.

Let $(A_n)_{n \in \Bbb{N}}$ be a sequence of pairwise distinct (not necessarily disjoint) sets in $S$.

For an arbitrary subset $M \subset X$ let $M^1 := M$ and $M^{-1} := M^c$.

For a sequence $\omega = (\omega_n)_{n} \in \{\pm 1\}^\Bbb{N}$, define

$$ B^\omega := \bigcap_{n \in \Bbb{N}} A_n^{\omega_n}. $$

Observe that $B^\omega \in A$, because it is a countable intersection of elements of $A$.

ALso note that $B^\omega \cap B^\gamma = \emptyset$ for $\gamma = (\gamma_n)_n \neq \omega$, because there is some $n$ such that $\gamma_n \neq \omega_n$, so that

$$ B^\omega \cap B^\gamma \subset A_n^{\omega_n} \cap A_n^{\gamma_n} = \emptyset. $$

Finally, note that for $n \in \Bbb{N}$ arbitrary, we have (why?)

$$ A_n = \bigcup_{\omega \text{ with } \omega_n = 1} B^\omega . $$

Hence, if the set $\{B^\omega \mid \omega \in \{\pm 1\}^\Bbb{N} \}$ were finite, it would easily follow that there could only be finitely many distinct $A_n$, a contradiction.

Hence, there is an infinite family $\omega^{(n)}$ with $B^{\omega^{(n)}} \neq B^{\omega^{(m)}}$ for $n \neq m$.

Let $C_n := B^{\omega^{(n)}}$. By discarding (at most) one set, we can assume $C_n \neq \emptyset$ for all $n$. Using the pairwise disjointness of the $C_n \neq \emptyset$, it is now easy to see that the map

$$ \Gamma : \{0,1\}^\Bbb{N} \to A, (\alpha_n) \mapsto \biguplus_{n \text{ with } \alpha_n = 1} C_n $$

is injective, because

$$ \Theta : A \to \{0,1\}^\Bbb{N}, M \mapsto \left(n \mapsto \begin{cases} 1, & \text{if }M\cap C_{n}\neq\emptyset\\ 0, & \text{if }M\cap C_{n}=\emptyset \end{cases} \right) $$

is a left inverse for $\Gamma$.

Answer 3

The proof above is terse. I've filled out the missing bits, any error corrections are greatly appreciated.

What we want to prove: A sigma algebra $\mathcal{F}$ of set $X$ is either finite or uncountable.

Proof: First suppose $X$ is finite. Then, the largest sigma algebra of $X$, $\mathcal{P}(X)$, is finite. Now, suppose $X$ is infinite. We assume that $\mathcal{F}$ is countably infinite, and derive a contradiction.

As $\mathcal{F}$ is countably infinite, denote $\mathcal{F} = \{A_i : i \in \mathbb{N} \}$. For $\forall x \in X$, define $B_x := \cap_{x \in A_i} A_i$. As $\mathcal{F}$ is countable, note that $B_x$ is a countable intersection of elements of $\mathcal{F}$, so is an element of $\mathcal{F}$. Therefore, for $\forall x, y,z \in X$, $B_x \cap B_y \in \mathcal{F}$, and $B_x \backslash B_z \in \mathcal{F}$.

I claim that $B_x \cap B_y \neq \emptyset \implies B_x = B_y$. As $B_x \cap B_y \in \mathcal{F}$, $\exists i_0$ such that $B_x \cap B_y = A_{i_0}$. Let $z \in B_x \cap B_y$. Then, $$B_z = \cap_{z \in A_i} A_i = \left( \cap_{\substack{z \in A_i \\ i\neq i_0}} A_i\right) \cap A_{i_0} \subset A_{i_0} = B_x \cap B_y.$$ Suppose that $ x \notin B_z$. Then, $x \in B_x \backslash B_z \in \mathcal{F}$. This is a contradiction because as $z \in B_x, B_z$, we know $B_x \backslash B_z$ is strictly smaller than $B_x$. However, $B_x$ is defined to be the intersection of all sets in $\mathcal{F}$ containing $x$, so is the `smallest' set in $\mathcal{F}$ containing $x$ which is also a subset of all other sets in $\mathcal{F}$ containing $x$. Therefore, we conclude $x \in B_z$. As $B_z$ also contains $x$, we obtain $B_x \subset B_z$. Thus, we have $$ B_x \subset B_z \subset B_x \cap B_y \subset B_x. $$ We can conclude $B_x = B_z = B_x \cap B_y$. By symmetry, $B_y = B_z = B_x \cap B_y$. Thus, $B_x = B_y$, and our claim is proven.

Let $B = \{ B_x\}_{x \in X}$. If $|B| = n < \infty$, then we have that $\mathcal{F}$ is finite. This is because for any $A \in \mathcal{F}$, $A$ can be expressed $A = \cup_{a \in A} B_a$ (as $B_a \subset A,$ $\forall a \in A$, and $B_x$ exists and is non-empty for $\forall x \in X$ as $x \in X \in \mathcal{F}$). So, there can exist at most $2^n$ elements of $\mathcal{F}$, a contradiction to the assumption that $\mathcal{F}$ is countably infinite. Now assume that $B$ is uncountably infinite. As we established $B_x \in \mathcal{F}, \forall x \in X$ (so $B \subset \mathcal{F}$), we have that $\mathcal{F}$ is uncountably infinite, a contradiction to the assumption that $\mathcal{F}$ is countably infinite.

Finally, assume that $B$ is countably infinite. Using the claim, we can let $B = \{ C_i \}_{i \in \mathbb{N}} \cup C_0$ where $C_0 = \emptyset$, and $C_i$ are disjoint non-empty sets such that $C_i = C_j \implies i = j$, $\forall i, j \in \mathbb{Z}_{\ge 1}$. Consider the set $C = \{\cup_{i \in I} C_i: I \subset \mathbb{N}\}$. Note that $I \neq J \implies \cup_{i \in I} C_i \neq \cup_{i \in J} C_i$. As $I$ runs over all elements of the power set of $\mathbb{N}$, $C$ has uncountably infinitely many elements. However, as each element of $C$ is composed of a countable union of elements of $\mathcal{F}$, $C \subset \mathcal{F}$. This is a contradiction to the assumption that $\mathcal{F}$ is countably infinite.

Answer 4

Proof by picture:

Take a collection of finitely many disjoint elements of the sigma algebra. Then the picture below shows gives some intuition as to why one can always add a element to this set of pairwise dijoint sets.

Once you have a infinite collection of pairwise disjoint sets one can identify each of these as distinct elements where unions of sets are also distinct. So by taking all countable unions on this collection one would generate a set with uncountably many elements as it is the power set of a countable set.

Answer 5

I elaborate ncmathsadist's answer a little bit:

We define the notion of atom: An atom is a non-empty minimal measurable set. That is, a non-empty measurable set $A$ is an atom iff for any measurable set $B\subseteq A$, either $B=\emptyset$ or $B=A$. Note that any two distinct atoms are disjoint. We have the following cases:

Case 1: There are finitely many atoms $A_{1},A_{2},\ldots,A_{n}$ and $\bigcup_{i}A_{i}=X$. In this case, the $\sigma$-algebra is finite and every measurable set is a union of atoms.

Case 2: There are finitely many atoms $A_{1},A_{2}\ldots,A_{n}$ but $\bigcup_{i}A_{i}\neq X$. (This includes the case that there is no atom) In this case, let $Y=X\setminus\bigcup_{i}A_{i}$. We can construct inductively a sequence of measurable set $\{B_{n}\}$ such that $Y=B_{1}\supsetneq B_{2}\supsetneq B_{2}\supsetneq\ldots$. Define $C_{1}=B_{1}\setminus B_{2}$, $C_{2}=B_{2}\setminus B_{3}$ etc, then $\{C_{n}\}$ is a sequence of pairwisely disjoint non-empty measurable sets.

Case 3: There are infinitely many atoms. Choose a countably infinite sub-family of atoms, then they are pairwisely disjoint.

Answer 6

Show that if a $\sigma$-algebra is infinite, that it contains a countably infinite collection of disjoint subsets. An immediate consequence is that the $\sigma$-algebra is uncountable.

Answer 7

Warning: the following proof is incorrect!

As ncmathsadist mentioned before, there is also a direct prove. For those who want to know how to proceed:

Let be $\mathcal{S}$ an infinite $\sigma$-Algebra. Take any pairwise distinct sets $A_1,A_2,\ldots\in\mathcal{S}$. Then $$\begin{align*}G_0&=A_0^\complement\cap A_1\cap A_2\cap A_3\cap\cdots,\\ G_1&=A_0\cap A_1^\complement\cap A_2\cap A_3\cap\cdots,\\ G_2&=A_0\cap A_1\cap A_2^\complement\cap A_3\cap\cdots,\\ \vdots\end{align*}$$ all lie in $\mathcal{S}$, because the countable intersection is closed in $\sigma$-Algebras, and it is easy to see, that they are pairwise disjoint. But also the countable and finite union of any collection of the $G_k’s$ must lie in $\mathcal{S}$.

You will recognise, that for distinct collections, you’ll get different unions (because the $G_k$’s are pairwise disjoint). Wich Collection you choose, you can encode with a binary sequence $a_0,a_1,a_2,\ldots$ by setting $a_k=1$ for taking $G_k$ into the collection and $a_k=0$ otherwise. So you get an injection from $\{0,1\}^\mathbb{N}$ to $\mathcal{S}$. But you already know, that $\{0,1\}^\mathbb{N}$ is an uncountable set. And that’s why $\mathcal{S}$ must also be uncountable.

Answer 8

Sigma algebras are just Venn diagrams. (with some caveats because of all the "countable union" business)

A sigma field $\mathcal{F}$ on $X$ defines an equivalence relation on $X$ where $x\sim y$ iff $\forall E\in \mathcal{F},x\in E\iff y\in E$. This partition is just the partition defined by the Venn diagram -- by the little intersection regions. The important point is that there is a bijection $\mathcal{F}\leftrightarrow \mathcal{P}(X/\sim)$ -- this should also be obvious with the Venn diagrams.

So what are the possible values for the cardinality of a power set?