Depending on what kit you have, accurately measuring run-out less than 0.01mm is quite a challenge. Dirt, surface imperfections, the instrument, and technique all add error. It may be that the lathe is slightly better or slightly worse than you think. Either way, it's pretty good.
I wouldn't worry about it unless the lathe is malfunctioning and – Hopper's point – you've eliminated all other sources of trouble first. For example, loose gibs and backlash are common maintenance / wear issues.
In the good old days operators didn't seem to worry about measuring their equipment. Professional lathes were periodically checked by specialists who had the necessary tools and training. Amateurs made do with what they had. Now we all have access to affordable DTI's etc it seems fashionable to check new machines out – I've done it myself.
I'm not convinced it's sensible; even in well-trained hands expensive measuring kit can mislead horribly. Bit like going on the internet with a sniffle and convincing yourself you must have Black-water Fever.
In practice, you can still get good results from 'faulty' equipment – if run-out really mattered, no-one would ever use a 3-jaw chuck. (Even a wonky chuck spins around a fixed axis, and for many jobs it's only the axis that matters.)
Nowadays I only take measurements off a machine when it doesn't cut as expected, or when resetting it. More often than not, faults are eye-ball obvious, like the top-slide backlash needs adjusting, or the tool height is shimmed wrong.
My experience is all with new kit. Are older machines that different, or are worn beds and shot bearings also eye-ball obvious?
Dave