19.6.1.2 Lebesgue measure and integration

Problem: 1) Find the Radon-Nikodym density with respect to the Lebesgue measure for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) 2) Find the Jordan expansion for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) What sets can be taken as \( X^{+}, X^{-} \)in Hahn decomposition??