in Old English a common prefix with nouns, adjectives, adverbs, and verbs, "out, outward, outer; forth, away," from out (adv.). The use was even more common in Middle English, and also with the senses "outer, outside, on the outside, from without, external, externally; apart; greatly, extremely; completely, thoroughly, to completion." Other senses of out that extended into the use as a prefix include "beyond the surface or limits; to the utmost degree; to an explicit resolution."
In composition out has either its ordinary adverbial sense, as in outcast, outcome, outlook, etc., or a prepositional force, as in outdoors, or forms transitive verbs denoting a going beyond or surpassing of the object of the verb, in doing the act expressed by the word to which it is prefixed, as in outrun, outshine, outvenom, etc. In the last use especially out may be used with almost any noun or verb. [Century Dictionary]
c. 1300, "an ordinance to fix the amount of a payment or tax," from Old French sise, shortened form of assise "session, assessment, regulation, manner," noun use of fem. past participle of asseoir "to cause to sit," from Latin assidere/adsidere "to sit beside" (and thus to assist in the office of a judge), "sit with in counsel or office," from ad "to" (see ad-) + sedere "to sit," from PIE root *sed- (1) "to sit."
Probably a misdivision of l'assise as la sise. The sense of "extent, amount, volume, magnitude" (c. 1300) is from the notion of regulating something by fixing the amount of it (weights, food portions, etc.). Specific sense of "set of dimensions of a manufactured article for sale" is attested from 1590s.