= Solution
<Collider luminosity> is the proportionality factor between a process <cross section> and its interaction rate: $dN/dt=\mathcal L\sigma$ before efficiency corrections. It has units of inverse area per time. <Integrated collider luminosity> is $\mathcal L_{\rm int}=\int\mathcal L(t)\,dt$, with inverse-area units, and gives $N=\sigma\mathcal L_{\rm int}$ for a constant <cross section> and unit efficiency.
At an $e^+e^-$ collider, count small-angle <Bhabha scattering> in a dedicated forward detector. Its calculable <quantum electrodynamics> <cross section> in a precisely defined angular acceptance provides $\mathcal L=N/(\epsilon\sigma\Delta t)$, after background and radiative corrections. The inner angular boundary is especially important because the rate is sharply forward-peaked.
At a <collider> with two <proton> beams, a <van der Meer scan> calibrates a visible interaction monitor by separating the beams horizontally and vertically while recording the overlap-dependent rate. Known bunch populations and the measured overlap give an absolute <collider luminosity> and hence the monitor's visible <cross section>. Its rate then tracks <collider luminosity> during normal running. Correlations between transverse coordinates require a nonfactorizing scan analysis rather than blindly multiplying two one-dimensional profiles.
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