Let $\mu$ be a measure on $X$ and $f$ be an integrable function. We define $\nu (E) = \int_{E} fd\mu$.
Show that $\nu_{+}(dx) = f_{+}(x)\mu(dx)$ and $\nu_{-}(dx) = f_{-}(x)\mu(dx)$, where $f_{+} = max(f, 0), f_{-} = max(-f, 0)$.
Moreover, show that $|\nu|(dx) = |f(x)|\mu(dx)$ and $|\nu|(X) = \int |f| d\mu$.
I don't even know how to start :( Any hints would be great!
Let $A = \{x : f(x) \ge 0\}$ and $B = \{x : f(x) < 0\}$. Show that $A$ is a $\nu$-positive set and $B$ is a $\nu$-negative set. You'll have, for every measurable set $E$, $\nu_+(E) = \nu(E\cap A) = \int_E f1_A\, d\mu = \int f_+\, d\mu$, and similarly $\nu_{-}(E) = -\nu(E\cap B) =\int_E -f1_B\, d\mu = \int_E f_{-}\, d\mu$. The first two parts follow from this. The third part follows from $\lvert \nu\rvert = \nu_+ + \nu_{-}$ and $\lvert f\rvert = f_+ + f_{-}$.