Suppose $(X, M, \mu)$ is a complete finite measure space with $f\in L^{1}(X, M, \mu)$. Define the signed measure $v(E) = \int_E f d\mu$.
Since we know that $\mu(E)<\infty$, do we also have that $-\infty< v(E)<\infty$ for all $E\in M$?
My thoughts are that, if $f$ ever achieved values of $-\infty$ or $\infty$, then we could have an infinite integral and hence the signed measure would not be finite for some set E ... however, I think this would contradict $f$ being $\mu$-integrable. I think that, if $f$ is integrable, then it should only have finite values ... then, since $f(x)$ is finite for all $x$ in consideration and $\mu(E)$ is finite for all $E$ in consideration, then the integral should be finite and hence the signed measure is finite.
Can someone clarify this for me? I have not been successful in finding an answer and so far, all I have to go on is my intuition (which is often times wrong). Links to additional resources are welcomed! Thank you in advanced!