Extension of Monotone Convergence Theorem and Dominated Convergence Theorem

Question

A very simple question: Do the Monotone Convergence Theorem and Dominated Convergence Theorems hold for signed measures? In particular, if $\mu$ is a signed measure and $f_{n} \to f$ pointwise a.e., is it true that $$ \lim_{n \to \infty} \int f_{n} d \mu= \int f d \mu $$ if the conditions of the either of the theorems are met? If so, how would one prove and extension of these theorems?

Answer 1

Let $X$ be the underlying space. Let $P, N$ be a Hahn decomposition of $X$. Let $\mu^{+}, \mu^{-}$ be the corresponding Jordan decomposition of $\mu$. Then $$ \int_{X} f_{n} d\mu = \int_{P} f_{n} d\mu^{+} - \int_{N} f_{n} d\mu^{-}. $$ You can then apply whatever theorem you like to each integral on the right-hand side separately.