The problem is Excercise 5. in page 88 of Folland's "real analysis: modern techniques and their applications", 2nd edition, as the image below shows.

As the hint indicates, we should use Excercise 4. From Excercise 4, if signed measure $\displaystyle \nu=\lambda-\mu$, then $\displaystyle |\nu|=\nu^++\nu^-\leq\lambda+\mu$. Supposing $\displaystyle \nu_1=\lambda_1-\mu_1$, $\displaystyle \nu_2=\lambda_2-\mu_2$, we have $\displaystyle \nu_1+\nu_2=(\lambda_1+\lambda_2)-(\mu_1+\mu_2)$. Then it follows $\displaystyle |\nu_1+\nu_2|\leq\lambda_1+\lambda_2+\mu_1+\mu_2$. But at the same time $\displaystyle |\nu_1|\leq\lambda_1+\mu_1$, $\displaystyle |\nu_2|\leq\lambda_2+\mu_2$, so $\displaystyle |\nu_1|+|\nu_2|$ also $\displaystyle \leq\lambda_1+\lambda_2+\mu_1+\mu_2$. I really have no idea how to employ Excercise 4. to prove $\displaystyle |\nu_1+\nu_2|\leq|\nu_1|+|\nu_2|$. Can you help me? Thanks in advance!