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Purpose of measure $\mu \equiv 0$. I just stumbled on the exclusion of this case in a proof and thought it was a funny measure. Does this measure have any purpose in anything?

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    Yes. It is useful for asking questions about. (And horrible grammar with badly used prepositions.) – 2017-01-05
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    I don't see any prepositions :p But yes, point taken. – 2017-01-05
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    It was a comment on my own grammar :-). – 2017-01-05
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    Haha, but all right then ;) – 2017-01-05
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    As an aside, 'extreme' cases such as $\mu=0$ are often a good test of understanding. – 2017-01-05
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    Indeed, I had to go back and check every proof so far since I hadn't even thought of it. – 2017-01-05
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    @copper.hat Ending sentences with prepositions has _never_ been improper. It's strictly meme among poorly educated elementary school marms. Ditto with split infinitives. – 2017-01-05
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    @B.Goddard: You mean all of my beatings were for nothing? – 2017-01-05
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    @copper.hat Only if you think "building character" is nothing;-) – 2017-01-05
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    @B.Goddard: Still hate the taste of carbolic soap. – 2017-01-05

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The set of finite signed measures on a $\sigma$-algebra is a vector space with respect to the usual operations. That vector space needs an identity element.