The k-Dimensional Borel $\sigma$-Field

For each $k \in \mathbb {N}$ and $j=1, \ldots , k$, every $\left(k-j\right)$-dimensional hyperplane is in $\mathscr {B}^{k}$.

Consider the circumference of the unit circle in $\mathbb {R}^{2}$, the set of points $\left(x, y\right)$ with $x^{2}+y^{2}=1$. The set of points on the circumference of the unit circle having at least one irrational coordinate is a two-dimensional Borel set.

The set of points in $\mathbb {R}^{k}$ having all transcendental coordinates is in $\mathscr {B}^{k}$.

For any rectangle [of any form] $A \subseteq \mathbb {R}^{k}$, we define the $k$-dimensional Borel $\sigma$-field over $A$ to be the $\sigma$-field generated by all $k$-dimensional rsc subrectangles of $A$; call this class $\mathscr {B}_{A}^{k}$. $\mathscr {B}_{A}^{k}$ may be equivalently defined as $\left\{ B \cap A: B \in \mathscr {B}^{k}\right\}$.

$\mathscr {B}^{k}$ is generated by a countable class.

Show that $\mathscr {B}^{k}$ is not generated by the singletons of $\mathbb {R}^{k}$, nor by the finite subsets of $\mathbb {R}^{k}$, nor by the amc subsets of $\mathbb {R}^{k}$.

Here are five extra definitions of $\bar{\mathscr {B}}^{k}$ that are found elsewhere. These five definitions are equivalent to our definition of $\bar{\mathscr {B}}^{k}$.

1. The class of subsets of $\overline{\mathbb {R}^{k}}$ that are $\mathscr {B}^{k}$-sets or are $\mathscr {B}^{k}$ sets augmented by points that have $\pm \infty$ in at least one coordinate.

2. The $\sigma$-field of subsets of $\overline{\mathbb {R}^{k}}$ generated by the class of open rectangles in $\overline{\mathbb {R}^{k}}$ of the form $(\boldsymbol {a}, \boldsymbol {b})$, $[-\infty , \boldsymbol {a})$, and $(\boldsymbol {a},+\infty ]$, where $\boldsymbol {a}, \boldsymbol {b} \in \mathbb {R}^{k}$.

3. The $\sigma$-field on $\overline{\mathbb {R}^{k}}$ generated by $\left\{ (-\infty , \boldsymbol {x}]: \boldsymbol {x} \in \overline{\mathbb {R}^{k}}\right\}$.

4. $\sigma \left(\mathscr {B}^{k} \cup \left\{ \left\{ \boldsymbol {x}\right\} : \text{each } x_{i} \text{ freely equals } +\infty \text{ or } -\infty \right\} \right)$.

5. The $\sigma$-field on $\overline{\mathbb {R}^{k}}$ generated by the open subsets of $\overline{\mathbb {R}^{k}}$, that is, the sets of the form $A$, $A \cup (\boldsymbol {x},+\infty ]$, $A \cup [-\infty , \boldsymbol {x})$, and $A \cup [-\infty , \boldsymbol {x}) \cup (\boldsymbol {y},+\infty ]$, where $A$ is an open subset of $\mathbb {R}^{k}$ and $\boldsymbol {x}, \boldsymbol {y} \in \mathbb {R}^{k}$.

Let $\Omega$ and $\Omega^{\prime }$ be sets, and let $\mathcal{F}$ and $\mathcal{F}^{\prime }$ denote fields over $\Omega$ and $\Omega^{\prime }$, respectively. Let $\mathcal{F} * \mathcal{F}^{\prime }$ denote the collection of all finite disjoint unions of sets of the form $A \times B$ where $A \in \mathcal{F}$ and $B \in \mathcal{F}^{\prime }$.