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?
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$\begingroup$ Every signed measure can be decomposed as $\mu=\mu_+-\mu_-$ where $\mu_{\pm}$ are (non-signed) measure... $\endgroup$– SurbCommented Nov 15, 2020 at 14:08
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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.
