The total variation of a measure is defined as $$|\mu|(E)=\sup \sum_i |\mu(E_i)|$$ where the supremum is taken over all countable partitions of $E$.
So the positive and negative variation of $\mu$ is $$\mu^{+}(E)=\dfrac{1}{2}\left(\left|\mu\right|(E) + \mu(E)\right) \qquad \mu^{-}(E)=\dfrac{1}{2}\left(\left|\mu\right|(E) - \mu(E)\right)$$
But I also saw the following definition for $\mu^{\pm}$, $$\mu^{+}(E)=\sup\{\mu(F) : F\in\mathcal{F},~F\subset E\} \qquad \mu^{-}(E)=-\inf\{\mu(F) : F\in\mathcal{F},~F\subset E\}$$
These two definitions must be equivalent, right? How to prove that they are equivalent?