Half-Open Rectangles and Coordinate Volume

Boxes and Coordinate Volume

Rectangles in \(\mathbb{R}^n\) provide a basic geometric language for describing size one coordinate at a time. Their half-open form, which includes the left endpoint of each coordinate interval and excludes the right endpoint, is particularly useful in measure theory. Here we define these boxes and assign them a coordinate volume. This assignment is a formula based on side lengths; the arguments below do not establish that it agrees with Lebesgue measure.

Fix an integer \(n\ge1\). Write \(\overline{\mathbb{R}}=[-\infty,\infty]\) for the extended real line. For \(a,b\in\overline{\mathbb{R}}\) with \(a\le b\), define the coordinate interval

[a,b)=\{x\in\mathbb{R}:a\le x<b\}.

The comparisons use the usual extended-real order. In particular, \([-\infty,b)=\{x\in\mathbb{R}:x<b\}\) for finite \(b\), \([a,\infty)=\{x\in\mathbb{R}:x\ge a\}\) for finite \(a\), and \([-\infty,\infty)=\mathbb{R}\). If \(a=b\), the interval is empty: no real \(x\) can satisfy both \(a\le x\) and \(x<a\). The intervals are Borel subsets of \(\mathbb{R}\), including the empty interval and \(\mathbb{R}\).

Definition: A half-open rectangle, or box, in \(\mathbb{R}^n\) is specified by endpoint vectors \(a=(a_1,\ldots,a_n)\) and \(b=(b_1,\ldots,b_n)\) in \(\overline{\mathbb{R}}^n\), with \(a_i\le b_i\) for every \(i\), and is the set \(R=\prod_{i=1}^n[a_i,b_i)\). Its \(i\)th side length is \(\ell_i=0\) if \(a_i=b_i\); otherwise \(\ell_i=b_i-a_i\) when both endpoints are finite, and \(\ell_i=\infty\) if \(a_i=-\infty\) or \(b_i=\infty\). Its coordinate volume \(V(R)\) is \(0\) if any \(\ell_i=0\); if every \(\ell_i>0\), it is the product \(\prod_{i=1}^n\ell_i\), interpreted as \(\infty\) if at least one factor is \(\infty\).

The zero-side convention matters: the product notation for extended nonnegative numbers does not assign a value to \(0\cdot\infty\). We first check whether any side is zero. If so, the box is empty and its assigned volume is zero. If every side is positive, the product is unambiguous: it is finite when all sides are finite, and infinite when at least one side is infinite. For a nonempty box, the endpoint data in each coordinate interval are determined by that interval; for an empty box, different endpoint data can describe the same empty set, but the volume is always zero.

Calculating Volumes and Checking Membership

Worked Example: A Finite Three-Dimensional Box

Let \(R=[-2,1)\times[0,4)\times[3,5)\subset\mathbb{R}^3\). The three side lengths, computed by subtracting the left endpoint from the right endpoint, are \(1-(-2)=3\), \(4-0=4\), and \(5-3=2\). None is zero or infinite, so

V(R)=3\cdot4\cdot2=24.

For membership, a point \((x_1,x_2,x_3)\) belongs to \(R\) exactly when \(-2\le x_1<1\), \(0\le x_2<4\), and \(3\le x_3<5\). Thus \((-2,0,3)\in R\), since all three left endpoints are included. In contrast, \((1,2,4)\notin R\), because the first coordinate equals the excluded right endpoint \(1\). Check: the side-length product is \(24\), and the corner test confirms the left-inclusive, right-exclusive convention in every coordinate.

Worked Example: A Degenerate Box

Consider \(D=[1,1)\times[-2,3)\subset\mathbb{R}^2\). The first interval is empty because it would require \(1\le x<1\), so \(D=\varnothing\). Its side lengths are \(\ell_1=0\) and \(\ell_2=3-(-2)=5\). By the zero-side convention, \(V(D)=0\).

Now consider \(D'=[1,1)\times[-\infty,\infty)\). Its side lengths are \(0\) and \(\infty\), respectively. The set is still empty, and \(V(D')=0\), not an undefined product. Check: in both boxes the first coordinate condition is impossible, so both sets are empty and the assigned volumes are zero.

Worked Example: Two Unbounded Coordinates

Let \(S=[-\infty,0)\times[2,\infty)\times[-1,1)\subset\mathbb{R}^3\). The first coordinate interval is \((-\infty,0)\), and its length is \(\infty\). The second interval is \([2,\infty)\), also of length \(\infty\). The third interval has length \(1-(-1)=2\). All three side lengths are positive, and two are infinite, so

V(S)=\infty\cdot\infty\cdot2=\infty.

For example, \((-2,2,0)\in S\): \(-2<0\), \(2\ge2\), and \(-1\le0<1\). But \((0,3,0)\notin S\), because the first interval excludes \(0\). Check: both of the first two coordinate intervals are unbounded, and neither has equal endpoints; hence neither gives a zero side that could make the box empty.

Worked Example: One Unbounded Coordinate

Let \(T=[-3,1)\times[4,\infty)\times[2,6)\subset\mathbb{R}^3\). The side lengths are \(1-(-3)=4\), \(\infty\), and \(6-2=4\). Since there is no zero side and one side is infinite, \(V(T)=4\cdot\infty\cdot4=\infty\).

The point \((-3,4,2)\) belongs to \(T\), since each left endpoint is included, including the finite left endpoint of the unbounded interval. The point \((0,4,6)\) does not belong to \(T\), because \(6\) is excluded from the third coordinate interval. Check: replacing the last coordinate \(6\) by \(5\) gives \((0,4,5)\in T\); the only membership failure in \((0,4,6)\) is its third coordinate.

Two Properties of the Assignment

Proposition: For a half-open rectangle \(R=\prod_{i=1}^n[a_i,b_i)\), \(V(R)=0\) if and only if at least one coordinate interval is empty. If no coordinate interval is empty, then \(V(R)>0\), with \(+\infty\) allowed as a positive value.

Proof. If some coordinate interval is empty, then its endpoints are equal and its side length is zero. The definition gives \(V(R)=0\). Conversely, suppose no coordinate interval is empty. Then \(a_i<b_i\) for every \(i\). If the endpoints are finite, their difference \(b_i-a_i\) is strictly positive. If an endpoint is infinite, the side length is \(\infty\), which is also positive. Thus every side length is positive. If they are all finite, their finite product is positive; if one or more are infinite, the defined volume is \(\infty\). In either case \(V(R)>0\). \(\square\)

Proposition: If \(R\) and \(S\) are half-open rectangles in \(\mathbb{R}^n\) and \(R\subseteq S\), then \(V(R)\le V(S)\), using the usual order on \([0,\infty]\).

Proof. If \(R\) is empty, then \(V(R)=0\), so \(V(R)\le V(S)\) because every coordinate volume is nonnegative. Suppose instead that \(R\) is nonempty. For each coordinate \(i\), let \(I_i\) and \(J_i\) be the \(i\)th coordinate intervals of \(R\) and \(S\). For any \(x_i\in I_i\), nonemptiness of every coordinate interval of \(R\) allows us to choose one point from each of its other coordinate intervals. Together these coordinates form a point in \(R\), and therefore in \(S\). Its \(i\)th coordinate is \(x_i\), so \(x_i\in J_i\). This proves \(I_i\subseteq J_i\) for every \(i\). Consequently the side length of \(I_i\) is no greater than that of \(J_i\): containment cannot increase a finite interval’s length, and if the smaller interval has infinite length, the containing interval also has infinite length. Neither box has an empty coordinate interval, so all its side lengths are positive. If some side of \(S\) is infinite, \(V(S)=\infty\) and the inequality follows. Otherwise every side of \(S\), and hence of \(R\), is finite; multiplying the coordinatewise inequalities between these nonnegative finite lengths gives \(V(R)\le V(S)\). \(\square\)

Common Pitfalls and Scope

Pitfall: multiplying before handling a degenerate side. For \(D'=[1,1)\times[-\infty,\infty)\), a tempting but invalid calculation is \(0\cdot\infty=\text{undefined}\), followed by the conclusion that the volume is undefined. The correction is to use the definition in its stated order: the first side length is zero, so \(D'\) is empty and \(V(D')=0\). The infinite length of another coordinate does not change that assignment.

Pitfall: including the right endpoint. The condition \(x\in[a,b)\) is \(a\le x<b\), not \(a\le x\le b\). Thus \(1\notin[-2,1)\), while \(-2\in[-2,1)\). Ignoring this distinction changes the box as a set, even though the endpoint convention is built into every coordinate.

Coordinate volume is a geometric assignment made from endpoint data. In particular, the calculations here do not prove additivity under subdivisions or identify \(V(R)\) with \(m(R)\), where \(m\) denotes Lebesgue measure on \((\mathbb{R}^n,\mathcal{L})\). Lebesgue measure is complete and sigma-finite; establishing its relationship to this coordinate formula requires further results. The inclusion property proved above is a direct consequence of the formula and does not require a measure-theoretic claim.

Check Your Understanding

For each question, keep the left-included, right-excluded convention and the zero-side rule in view.

Compute the coordinate volume of \([0,3)\times[-2,2)\) and state whether \((3,0)\) belongs to the box.
What is the volume of \([-1,1)\times[4,4)\times[0,\infty)\)? Explain why the infinite side does not make the answer ambiguous.
For \(Q=[-\infty,2)\times[1,5)\), calculate its coordinate volume and decide whether \((2,3)\) belongs to \(Q\).
Prove directly from the interval definition that \([a,b)\) is empty when \(a=b\).
If \(R\subseteq S\) and \(R\) is nonempty, why must each coordinate interval of \(R\) be contained in the corresponding coordinate interval of \(S\)?