To accurately measure the diameter of a wire, a micrometer or vernier calliper is used. Most micrometers and verniers have a digital read out.
Both the micrometer and the vernier are suitable instruments with which to measure these diameters, although a micrometer would be preferable for diameters smaller than about 1mm.
When it comes to fine wire mesh, the vernier is not up to the job. Stainless Steel Wire and Mesh supplies mesh with wire diameters from 0.025mm diameter to 3.15mm diameter. Clearly, a measuring instrument with a resolution of 0.01mm is not suitable for measuring a wire diameter of 0.025mm, and a micrometer must be used.
We suggest that a micrometer should be used for wire diameters of 1mm or less.