TITLE: Lebesgue integral and absolute value
QUESTION [9 upvotes]: I wonder why we say that $f$ is integrable iff $\int|f|\,d\mu$ is finite? Why we use absolute value? Won't it be enough to have that $\int f\, d\mu$ is finite to call $f$ integrable? Are there functions $f$ for which $\int f \ d\mu$ is finite but $\int |f| \ d\mu$ is not (like conditional convergence for series)?. But then why we would call such $f$ not Lebesgue integrable?

REPLY [14 votes]: As far as why we define integrability this way, one reason that I can think of is that we cannot always assign a numerical value to $\int f d\mu$, but with $\int |f| d\mu$, we can - it's either finite or infinite. That is, $\int f d\mu$ is not always well-defined. However, if we know that $\int |f| d\mu < \infty$, then $\int f d\mu$ is well-defined since 
$$\int |f| d\mu = \int f^+ d\mu + \int f^- d\mu < \infty$$ which is true only when both positive and negative parts of the function are both finite. Thus,
$$\int f d\mu = \int f^+ d\mu - \int f^- d\mu $$ is finite as well.