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Let $\mu$ a signed measure, and $f$ a integrable function with respect to $|\mu|$, and if $\nu$ is defined for every mensurable set by $$\nu(E) = \int_E f \, d\mu$$

then $\nu \ll \mu$.

I need some tips, because in the exercise I have a hint, that is:

$$\int f \, d\mu = \int f \, d\mu^+ - \int f \, d\mu^-$$

If $\mu$ is a finite signed measure, then, for every measurable set $E$

$$|\mu|(E) = \sup \left|\int_E f \, d\mu \right|$$ where the supremum is extended over all measurable functions $f$ such that$|f| \leq 1$.

In the case that $\mu$ is finite sigbed measure is it solve right? Or there is some things that I forget? Thanks so much

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