What is the difference between the nouns measurement and measuring? Can I say the measurement has stopped the same way I can say the measuring has stopped?
Syntactically, you have different complements available. The verb measuring simply takes a direct object. The noun measurement and the gerundial noun measuring take PP complements headed by of. Thus you have:
Measuring the elephant was dangerous. (verb)
The measuring of the elephant was fraught with danger. (gerundial noun)
The measurement of the elephant went surprisingly well. (noun)
For one, measurement is a noun, measuring is a gerund.
"the measurement has stopped" refers to the process called measurement.
"the measuring has stopped" refers to the action of measuring in a modified way, which is why that part of speech is called a gerund.
Both sentences convey the same meaning. However, one refers to an object and the other indirectly to an action.