Example. Volumes of four-coordinate boxes [fdip-004D]

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Keep the four-cycle carrier \(V=\{0,1,2,3\}\). For real \(a_i\le b_i\), let \[B=\{x\in \mathbb R^4:a_i<x_i\le b_i\text { for }i=0,1,2,3\}.\] The coordinate Lebesgue measure gives \[\lambda _V(B)=\prod _{i=0}^3(b_i-a_i).\] Each factor is the length of its half-open interval, and the defining rectangle law multiplies them. If an endpoint pair is equal, the box is empty and the formula gives zero.

If every lower endpoint is zero and every upper endpoint is one, the box has volume \(1^4=1\). For lower endpoints \((0,0,-1,0)\) and upper endpoints \((2,1,1,3)\), its side lengths are \((2,1,2,3)\), so \[\lambda _V(B)=2\cdot 1\cdot 2\cdot 3=12.\] For \(a_i=-k\) and \(b_i=k\) in all 4 coordinates, with \(k>0\), the volume is \((2k)^4\). These volumes are unbounded as \(k\) increases, so the whole configuration space has infinite volume. Neither the edge weights nor a mass parameter enter these values. Reference volume is not a probability law; the later Gaussian density supplies a normalized weighting.

This is the rectangle formula of Definition [fdip-002H], from [tao2011measure, Proposition 1.7.11, pp. 197-199, preliminary version]. Mathlib's MeasureTheory.Measure.pi_pi gives the finite-product rectangle law; Real.volume_Ioc gives each interval length.