Following up on user2922582's answer, I note that The American Heritage Dictionary of the English Language, fifth edition (2011) has the same two definitions for remit as a noun that appear in the online AHDEL:
remit ... n. 1. A matter remitted for further consideration. 2. Chiefly British An area of responsibility; scope.
Merriam-Webster Online, too, has taken notice of the British English sense of the term:
remit noun 1 British : an area of responsibility or authority —usually singular [The problem was outside/beyond our remit.] [The BBC simply no longer sees producing Shakespeare as part of its remit as a public service television broadcaster. —John Morrison] 2 : an act of remitting 3 : something remitted to another person or authority
These appear to be relatively recent developments, however. For example, The American Heritage Dictionary of the English Language, fourth edition (2001) has a significntly different pair of entries for remit as a noun:
remit ... n. 1. The act of remitting, especially the referral of a case to another court. 2. A matter remitted for further consideration.
And two other U.S. dictionaries of similar vintage show similar ignorance of (or disregard for) remit in the "chiefly British" sense. From Encarta World English Dictionary (1999):
remit ... n. 1. LAW TRANSFER OF LEGAL CASE the transfer of a legal case from a higher to a lower court for further action to be taken 2. SOMETHING REMITTED something sent to another person or authority for consideration
(I might mention here that in the many U.S. judicial opinions I read during three years of law school, the usual term for the act of returning a case to a lower court for further action was remand, not remit.) And from Merriam-Webster's Eleventh Collegiate Dictionary (2003):
remit n (15c) 1 : an act of remitting 2 : something remitted to another person or authority
My copy of the massive and unabridged Webster's Third New International Dictionary (1961/1986) has an even narrower definition of the noun remit:
remit n an act of remitting or a matter, cause, or proceeding remitted to to another person or authority for consideration or judgment
The Oxford American Dictionary and Thesaurus (2003) comes closer to including the AHDEL's "chiefly British" meaning at a somewhat earlier date, although the wording of the first definition is so vague that I am unsure how close it actually is to that meaning:
remit ... n. 1 the terms of reference of a committee, etc. 2 an item remitted for consideration.
If "terms of reference" in definition 1 of this entry means something like "claimed ambit of responsibility," then this definitions is quite close to AHDEL 5's definition 2—but it isn't clear to me that the phrase has that particular meaning.
Even 15 years ago, U.S. English dictionaries showed very little awareness of remit in the sense of "an area of responsibility." That has changed within the past decade, but even now, U.S. usage of remit in the chiefly British sense does not appear to be widespread. And with the rise of "beyond my pay grade" as an alternative to saying "not within my remit," there doesn't seem to be much of a groundswell in popular U.S. usage toward increased adoption of remit to mean "purview, scope, or area of responsibility."