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This question is similar to this, but I didn't understand the answer.

In Wikipedia, is said that

If the measure $\mu$ is a complex measure, its upper and lower variation cannot be defined.

Why is it necessary to define in a different way the total variation of a complex measure? Are the two definitions equivalent if it is a signed measure?

Mrcrg
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1 Answers1

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There is no notion of inequalities for complex numbers. So the notions of supremum and infimum don't exist in $\mathbb C$ and upper and lower variation cannot be defined for complex measures.

For a signed measure the two notions are equivalent. Ref: Rudin's RCA.

Kavi Rama Murthy
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