Measuring width is quite easy using any caliper which reads to 0.1mm. At the machine, I use a carbon fiber dial caliper, which can’t damage the stone even when I’m clumsy.
Girdle thickness is harder. It can be measured directly with an instrument like the Bausch & Lomb Lenscope with a reticule, but that’s relatively expensive and awkward to use at the machine. The best way I know is a comparison with a thin wire or other object near the desired width. I’ve found that a jeweler’s saw blade works well for most girdles. Saw blades are in various thicknesses, so you’ll have to use a micrometer to measure them. You can also use a length of wire – there’s a useful chart at AWG to mm