c. 1200, "moderation, temperance, abstemiousness;" c. 1300, "instrument for measuring," from Old French mesure "limit, boundary; quantity, dimension; occasion, time" (12c.), from Latin mensura "a measuring, a measurement; thing to measure by," from mensus, past participle of metiri "to measure," from PIE root *me- (2) "to measure." The native word was Old English cognate mæð "measure."
Meaning "size or quantity as ascertained by measuring" is from early 14c. Meanings "action of measuring; standard measure of quantity; system of measuring; appointed or allotted amount of anything" are from late 14c. Also from late 14c. are the senses "proper proportion; balance." Sense of "that to which something is compared to determine its quantity" is from 1570s.
In music, from late 14c. as "air, tune;" 1570s as "rhythmic pattern." Specifically as "a group of tones indicated between two primary beats" is from 1660s. From mid-15c. as "rhythmic pattern in poetry;" c. 1500 in dance. Meaning "treatment 'meted out' to someone" is from 1590s; that of "plan or course of action intended to obtain some goal" is from 1690s; sense of "legislative enactment" is from 1759. Figurative phrase for good measure is from good measure as "ample in quantity in goods sold by measure" (late 14c.).