----- > [!proposition] Proposition. ([[products and Borel σ-algebras commute for second-countable spaces]]) > If $X$ and $Y$ are [[topological space|topological spaces]], then the [[Borel set|Borel σ-algebra]] on the [[product topology|product space]] $X \times Y$ contains the [[product of σ-algebras|product]] of Borel [[σ-algebra|σ-algebras]] on the factors $X$ and $Y$: $\mathcal{B}(X \times Y) \supset \mathcal{B}(X) \otimes \mathcal{B}(Y).$ If $X$ and[^1] $Y$ are in fact [[second-countable space|second-countable]], then the reverse inclusion holds: $\mathcal{B}(X \times Y)=\mathcal{B}(X) \otimes \mathcal{B}(Y) \text{ when 2nd-countable}.$ > > ^proposition [^1]: In fact, I think the result only needs one of the factors to be [[second-countable space|second-countable]]. > [!proof]- Proof. ([[products and Borel σ-algebras commute for second-countable spaces]]) > By definition, $\mathcal{B}(X) \otimes \mathcal{B}(Y)$ is the smallest $\sigma$-algebra containing $\mathscr{A}=\{ A \times B : A \in \mathcal{B}(X), B \in \mathcal{B}(Y) \}$. Meanwhile, $\mathcal{B}(X \times Y)$ is the smallest $\sigma$-algebra containing $\{ U \times V: U \text{ open in }X , V \text{ open in }Y\}$. We'll show the two inclusions by showing that each contains the others generators. > > The first inclusion does not require second-countability: consider $A \times B$ with $A \in \mathcal{B}(X)$ and $B \in \mathcal{B}(Y)$. We have $A \times B=\pi ^{-1}(A) \cap \pi ^{-1} (B)$ where $\pi$ is [[measurable function|Borel measurable]] (it is [[continuous]]). Hence $A \times B$ is Borel in $X \times Y$ as an intersection thereof, $A \times B \in \mathcal{B}(X \times Y)$. Since $A \times B$ was arbitrary, we conclude $\mathcal{B}(X \times Y)$ contains $\mathscr{A}$, hence contains $\mathcal{B}(X) \otimes \mathcal{B}(Y)$. > > > For the reverse inclusion, let's now assume $X$ and $Y$ are [[second-countable space|second-countable]]. This implies $X \times Y$ is [[second-countable space|second-countable]]. By definition, $\mathcal{B}(X \times Y)$ is the smallest $\sigma$-algebra containing all the open sets in the [[product topology]] on $X \times Y$. By [[second-countable space|second-countability]] of $X \times Y$, any such open set is a [[countably infinite|countable]] union of basic open sets, and therefore is contained in $\mathcal{B}(X) \otimes \mathcal{B}(Y)$. Since $\mathcal{B}(X) \otimes \mathcal{B}(Y)$ contains all generating sets of $\mathcal{B}(X \times Y)$, it contains $\mathcal{B}(X \times Y)$ (the [[σ-algebra generated by a set collection|smallest σ-algebra]] satisfying such a property). > ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```