Half-Open Rectangles Closed under Intersection
Theorem
Let $\horectr {\mathbf a} {\mathbf b}$ and $\horectr {\mathbf c} {\mathbf d}$ be half-open $n$-rectangles.
Then $\horectr {\mathbf a} {\mathbf b} \cap \horectr {\mathbf c} {\mathbf d}$ is also a half-open $n$-rectangle.
Proof
As $\O$ is trivially a half-open $n$-rectangle, let us assume that:
- $\horectr {\mathbf a} {\mathbf b} \cap \horectr {\mathbf c} {\mathbf d} \ne \O$
By Cartesian Product of Intersections: General Case, it follows that:
- $\ds \horectr {\mathbf a} {\mathbf b} \cap \horectr {\mathbf c} {\mathbf d} = \prod_{i \mathop = 1}^n \hointr {a_i} {b_i} \cap \hointr {c_i} {d_i}$
which leaves only to prove that the $\hointr {a_i} {b_i} \cap \hointr {c_i} {d_i}$ are half-open intervals.
Now let $x \in \hointr {a_i} {b_i} \cap \hointr {c_i} {d_i}$.
Then $x$ is subject to:
- $x \ge a_i$ and $x \ge c_i$, i.e., $x \ge \max \set {a_i, c_i}$
- $x < b_i$ and $x < d_i$, i.e., $x < \min \set {b_i, d_i}$
and we see that these conditions are satisfied if and only if:
- $x \in \hointr {\max \set {a_i, c_i} } {\min \set {b_i, d_i} }$
Thus, we conclude:
- $\hointr {a_i} {b_i} \cap \hointr {c_i} {d_i} = \hointr {\max \set {a_i, c_i} } {\min \set {b_i, d_i} }$
showing that indeed the intersection is a half-open interval.
Combining this with the above reasoning, it follows that indeed:
- $\horectr {\mathbf a} {\mathbf b} \cap \horectr {\mathbf c} {\mathbf d}$
is again a half-open $n$-rectangle.
$\blacksquare$
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $\S 5$: Problem $7$