I don't know how to define the integral of a function with respect to a measure space $ (\Omega, \Sigma, \mu) $ where $ \mu $ is a signed measure $ \sigma$-finite in $ (\Omega, \Sigma) $.
Well, what I think is how $\mu $ is a signed measure $ \sigma $ -finite I can decompose it in the form $ \mu = \mu _ + - \mu _- $ where $ \mu _ +, \mu _- $ are positive measures, at least one of the two finites. So we define the integral of $ f $ with respect to $ \mu $ as $$\int_ {\Omega} fd\mu: = \int_{\Omega}fd\mu _ + - \int_{\Omega}fd\mu_-$$ and we say that $ f $ is integrable with respect to $ \mu $ if f is integrable with respect to $ \mu_+ $ and with respect to $ \mu_- $
Is this correct?