Exercise 0.15 (Product Borel $\sigma$-algebra)

These assertions could be viewed as a direct consequence of Exercise 1.4.17 and Exercise 1.4.18 of An Introduction to Measure Theory.

We give an alternative proof. We have to demonstrate that

By definition from Example 14 of the product ${\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}B(ℝ)$ of $n$ copies of $B$, and by definition from the Example 10 of Borel $\sigma$-algebra, we need to demonstrate that:

Recall the hint from the related Exercise 1.4.14 of Prof. Tao’s Measure Theory saying that to show that two families $F,F^{\prime }$ of sets generate the same $\sigma$-algebra, it suffices to show that every $\sigma$-algebra that contains $F$, contains $F^{\prime }$ also, and conversely. We make use of that trick.

• $B\left({ℝ}^n\right)\subseteq {\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}B(ℝ).$
  Let $A$ be an arbitrary $\sigma$-algebra containing $\left\{\left\{x\in {ℝ}^n:x_i\in E\right\}\in P\left(ℝ\right)|E\in B\left(ℝ\right)\&1\le i\le n\right\}$. To show that $A$ contains $O\left({ℝ}^n\right)$, let $E$ be an arbitrary open set of the Euclidean space. By Theorem 4.1 of Gamelin’s Topology, the open sets in ${ℝ}^n$ are the countable unions of product sets ${\bigcup <!--\; FUNCTION\; APPLICATION\; -->}_{\alpha }{E}_1^{\alpha }\times \dots \times E_n^{\alpha }$, where $E_i^{\alpha }$ are open subsets of $ℝ$ and therefore Borel measurable. We then have $E_1,\dots ,E_n\in B\left(ℝ\right)$. Therefore, $E_1\times ℝ\times \dots \times ℝ$, $ℝ\times E_2\times \dots \times ℝ$, $\text{…}$, $ℝ\times ℝ\times \dots E_n$ are contained in the base set $\left\{\left\{x\in {ℝ}^n:x_i\in E\right\}\in P\left(ℝ\right)|E\in B\left(ℝ\right)\&1\le i\le n\right\}$, and therefore their intersections and unions

  $$E=\bigcup _{\alpha }\left[\left(E_1^{\alpha }\times ℝ\times \dots \times ℝ\right)\bigcap \left(ℝ\times \underset{2}{\overset{\alpha }{E}}\times \dots \times ℝ\right)\bigcap \dots \bigcap \left(ℝ\times ℝ\times \dots \underset{n}{\overset{\alpha }{E}}\right)\right]$$

  must also be contained in $A$. Since every $\sigma$-algebra $A$ containing the cylinder set also contains the open sets of ${ℝ}^n$, the finest $\sigma$-algebra ${\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}B(ℝ)$ containing the cylinder set must be a $\sigma$-algebra containing the open sets of ${ℝ}^n$ as well.

• ${\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}B(ℝ)\subseteq B\left({ℝ}^n\right).$
  Let $E$ be an arbitrary set from the cylinder set, which generates ${\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}B(ℝ)$. Then $E$ is of the form $ℝ\times \dots \times E_j\times \dots \times ℝ$ for $1\le j\le n$ (by that we mean that $E_j$ is the $j$-th multiplicity in the Cartesian product) and $E_j\in B\left(ℝ\right)$. By Exercise 1.4.18 of An Introduction to Measure Theory, the product of Borel measurable sets is again Borel measurable in a product space, and so $E$ must be an element of the $\sigma$-algebra $B\left({ℝ}^n\right).$ Since ${\prod <!--\; FUNCTION\; APPLICATION\; -->}_{i=1}^{n}B(ℝ)$ is the finest $\sigma$-algebra containing $E$, it must be contained in $B\left({ℝ}^n\right).$