Let $f:X\to \Bbb C$ be a complex integrable function and $\mu$ a complex measure on $\sigma$-algebra generated by $X$. Is there any relation between $\displaystyle{\bf Re}\int fd\mu$ and $\displaystyle\int{\bf Re} (f) d\mu$ or $\displaystyle\int{\bf Re} (f) d|\mu|$?
Relation between integrals ${\bf Re}\int fd\mu$ and $\int{\bf Re} (f) d\mu$.
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measure-theory
1 Answers
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Always $$\displaystyle{\bf Re}\int fd\mu=\int{\bf Re} (f) d\mu$$
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0Also when $\mu$ is a complex measure? – 2017-02-01
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0@niki Yes: this follows directly by the definition. – 2017-02-01