Use your finger and thumb to feel both sides of the rim surfaces.
If it feels a bit concave, rather than flat, they're somewhat worn.
If the concave feel is quite 'deep' - time to replace the rims.
The above uses experience though; doing it for the first time you don't have any frame of reference. So pop by a couple of bike shops... I hope they won't just be interested in selling.
Accountantpete's advice is useful.
I measure 1.1 to 1.3mm with my lovely new dental gauge, so I feel reassured. Curious to know what the original thickness was though. Thanks again Andrew, the gauge was a great suggestion.
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