Exercise 1.7.19 (Examples of product $\sigma$-algebras)

Question

\begin{enumerate}[label=(oman*)]
\item Show that the product of two trivial $\sigma$-algebras (on two different spaces $X,Y$) is again trivial.
\item Show that the product of two finite $\sigma$-algebras is again finite.
\item Show that the product of two Borel $\sigma$-algebras (on two Euclidean spaces ${\bf R}^d, {\bf R}^{d'}$ with $d,d' \geq 1$) is again the Borel $\sigma$-algebra (on ${\bf R}^d 	imes {\bf R}^{d'} \equiv {\bf R}^{d+d'}$).
\item Show that the product of two Lebesgue $\sigma$-algebras (on two Euclidean spaces ${\bf R}^d, {\bf R}^{d'}$ with $d,d' \geq 1$) is not the Lebesgue $\sigma$-algebra.
\item However, show that the Lebesgue $\sigma$-algebra on ${\bf R}^{d+d'}$ is the completion of the product of the Lebesgue $\sigma$-algebras of ${\bf R}^d$ and ${\bf R}^{d'}$ with respect to $d+d'$-dimensional Lebesgue measure.
\item This part of the exercise is only for students who are comfortable with cardinal arithmetic. Give an example to show that the product of two discrete $\sigma$-algebras is not necessarily discrete.
\item On the other hand, show that the product of two discrete $\sigma$-algebras $2^X, 2^Y$ is again a discrete $\sigma$-algebra if at least one of the domains $X, Y$ is at most countably infinite.
\end{enumerate}

Elu Thingol · Accepted Answer

\begin{enumerate}[label=(oman*)]
\item 	extit{Product of two trivial $\sigma$-algebras (on two different spaces $X,Y$) is again trivial.}
ewline
  Using the generating set from the \href{./exercise_1-7-18}{previous exercise}, we case-by-case verify that
  $$\left\langle \left\{ \emptyset 	imes \emptyset, \emptyset 	imes Y, X 	imes \emptyset, X 	imes Y ight\} ightangle = \langle \{\emptyset, X 	imes Y\} angle.$$
  By \href{./exercise_1-4-9}{Exercise 1.4.9 (Recursive description of a generated Boolean algebra)}, we know that this equals to $\{\emptyset, X 	imes Y\}$, which is a trivial algebra.

\item 	extit{Product of two finite $\sigma$-algebras is again finite.}
ewline
  Let $n := \#\mathcal{B}_X$ and $m := \#\mathcal{B}_Y$. Then $\{ E 	imes F \mid E \in \mathcal{B}_X, F \in \mathcal{B}_Y\} \leq nm$ and we have
  $$\mathcal{B}_X 	imes \mathcal{B}_Y = \langle \{ E 	imes F \mid E \in \mathcal{B}_X, F \in \mathcal{B}_Y\} angle \subseteq 2^{\{ E 	imes F \mid E \in \mathcal{B}_X, F \in \mathcal{B}_Y\}}$$
  and the cardinality of the right hand side is less than or equal to $\leq 2^{nm}$.

\item 	extit{Product of two Borel $\sigma$-algebras (on two Euclidean spaces ${\bf R}^{d}, {\bf R}^{d'}$ with $d,d' \geq 1$) is again the Borel $\sigma$-algebra (on ${\bf R}^{d} 	imes {\bf R}^{d'} \equiv {\bf R}^{d+d'}$).}
ewline
  We demonstrate that $\mathcal{B}[\mathbf{R}^{d}] 	imes \mathcal{B}[\mathbf{R}^{d'}] = \mathcal{B}[\mathbf{R}^{d+d'}]$.
  \begin{itemize}
  \item[$\subseteq$] Let $E$ be a Borel measurable subset of $\mathbb{R}^{d}$, and let $F$ be a Borel measurable subset of $\mathbb{R}^{d'}$. By \href{./exercise_1-4-17}{Exercise 1.4.17}, $E 	imes F$ is a Borel measurable subset of $\mathbb{R}^{d+d'}$.
  \item[$\supseteq$] Let $E$ be a Borel measurable subset of $\mathbb{R}^{d+d'}$. By \href{./exercise_1-4-18}{Exercise 1.4.18}, for any $x_1\in\mathbb{R}^{d'}$, the slice $\{x_2\in\mathbb{R}^{d'}: (x_1,x_2)\in E\}$ is a Borel measurable subset of $\mathbb{R}^{d'}$.
  \end{itemize}
  Thus, every pair of sets in $\mathcal{B}[\mathbf{R}^{d}]$ and $\mathcal{B}[\mathbf{R}^{d'}]$ can be associated with a set in $\mathcal{B}[\mathbf{R}^{d+d'}]$, and vice versa.

\item 	extit{Product of two Lebesgue $\sigma$-algebras (on two Euclidean spaces ${\bf R}^d, {\bf R}^{d'}$ with $d,d' \geq 1$) is not the Lebesgue $\sigma$-algebra.}
ewline
  Consider two one-dimensional Lebesgue $\sigma$-algebras $\mathcal{L}[\mathbf{R}]$ and the Borel $\sigma$-algebra $\mathcal{L}[\mathbf{R}^2]$. Since a Lebesgue $\sigma$-algebra is always complete, it must in particular contain the null set $\{0\}	imes V$, where $V$ denotes the \href{./proposition_1-2-18}{Vitali set}. However, no two sets in $\mathbf{R}$ can result in this product since $V$ is non-measurable. Thus, we have found a set $\{0\}	imes V \in \mathcal{L}[\mathbf{R}^2]$ such that $\{0\}	imes V 
ot\in \mathcal{L}[\mathbf{R}]	imes\mathcal{L}[\mathbf{R}]$.

\item 	extit{The Lebesgue $\sigma$-algebra on ${\bf R}^{d+d'}$ is the completion of the product of the Lebesgue $\sigma$-algebras of ${\bf R}^d$ and ${\bf R}^{d'}$ with respect to $d+d'$-dimensional Lebesgue measure.}
ewline
\begin{itemize}
\item[$\subseteq$] Obviously $\mathcal{L}[\mathbb{R}^{d}]	imes\mathcal{L}[\mathbb{R}^{d'}] \subseteq \mathcal{L}[\mathbb{R}^{d+d'}]$ since the former is the coarsest $\sigma$-algebra on $\mathbb{R}^{d+d'}$ containing the product sets and $\mathcal{L}[\mathbb{R}^{d+d'}]$ contains all product sets by \href{./exercise_1-2-22}{Exercise 1.2.22}. Furthermore, \href{./exercise_1-4-27}{Lebesgue $\sigma$-algebra is complete}; thus, the completion of $\mathcal{L}[\mathbb{R}^{d}]	imes\mathcal{L}[\mathbb{R}^{d'}]$ must be contained in $\mathcal{L}[\mathbb{R}^{d+d'}]$ as well.
\item[$\supseteq$] Recall that by \href{./exercise_1-4-19}{Exercise 1.4.19}, the Lebesgue $\sigma$-algebra is generated by the union of the Borel $\sigma$-algebra and the Borell null-algebra. Thus, by part (iii) of this exercise, we obtain
$$\mathcal{L}[\mathbf{R}^{d+d'}] = \overline{\mathcal{B}[\mathbf{R}^{d+d'}]} = \overline{\mathcal{B}[\mathbf{R}^{d}]	imes \mathcal{B}[\mathbf{R}^{d'}]}.$$
But $\mathcal{B}[\mathbf{R}^{d}]	imes \mathcal{B}[\mathbf{R}^{d'}] \subseteq \mathcal{L}[\mathbf{R}^{d}]	imes \mathcal{L}[\mathbf{R}^{d'}]$, and so we are done.
\end{itemize}

\item 	extit{The product of two discrete $\sigma$-algebras is not necessarily discrete.}
ewline
	exttt{this part is yet to be solved by a student who is comfortable with cardinal arithmetic.}

\item 	extit{The product of two discrete $\sigma$-algebras $2^X, 2^Y$ is again a discrete $\sigma$-algebra if at least one of the domains $X, Y$ is at most countably infinite.}
ewline
	exttt{this part is yet to be solved by a student who is comfortable with cardinal arithmetic.}

\end{enumerate}

Exercise 1.7.19 (Examples of product $\sigma$-algebras)

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Exercise 1.7.19 (Examples of product $\sigma$-algebras)

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