The term `frequency' in EE (especially signal theory)
Generally, frequency is measured in Hz, or 1/second.  If a variable named "f" or "Fsomething" is used, then its unit is generally understood to be Hz.

Sometimes it is mathematically more convenient to use radians/second instead of full rotations per second.  If a variable is named "&omega;", then its unit is generally understood to be radians/second.

All that said, it's really sloppy not to specify the units.  Mostly you'll run into rad/s in texts and papers.  Look carefully at the beginning of a paper, the start of a chapter in a book or maybe in a definitions section, and you should find a definition of the terms used.  Or, something like your example of introducing the fundamental frequency as "&omega;0 = 2&pi;f0" makes it quite clear.  That basically defines how variables "&omega;" and "f" will be used subsequently, which is also the overwhelming convention.