MMA are commonly used to denote a large, metropolitan municipal entity.
Is there a difference between the two? Is there a certified definition to these terms? Google is a bit inconclusive about the issue.
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I'm assuming you're referring to specific to the USA. After much searching I came across this page: http://www.census.gov/population/www/documentation/twps0006/table03.html which states:
Major metropolitan areas (MMAs; areas of one million or more population)
I can't find anything official for DMA. It seems to be used unofficially for "District Metropolitan Area", specifically Washington DC usually but that's it.
Found the same definition for MMA as GIS-Jonathan. For DMA though, I found this at Wikipedia.
A media market, broadcast market, media region, designated market area (DMA), Television Market Area (FCC term), or simply market is a region where the population can receive the same (or similar) television and radio station offerings, and may also include other types of media including newspapers and Internet content. They can coincide or overlap with 1 or more metropolitan areas, though rural regions with few significant population centers can also be designated as markets.
Could be the MMA you're referring to.
In a Marketing context, DMA are Designated Market Areas.
DMA (Designated Market Area) regions are the geographic areas in the United States in which local television viewing is measured by The Nielsen Company. The DMA data are essential for any marketer, researcher, or organization seeking to utilize standardized geographic areas within their business.
The actual limits are updated every Fall by Nielsen, and there were about 210 of them last time I checked.
Here's one simple raster map of DMAs.