Originally it was measured by people like Lavoisier doing careful experiments measuring the ratios of the weights of elements reacting with each other. For instance the weight of mercury and the weight when it has all been converted to the oxide.
Nowadays, mass spectrometry is one of the most common techniques. That depends on bending a beam of ionized material (atoms or molecules) by a magnetic or electric field. The heavier the ion, the less it deviates. Other related technology can also be used.
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