# Role of positivity in integration by $f\in L^1$ iff $|f|\in L^1$.

This is related to Lang, Real and Functional Analysis pg 142, Chpt VI, Sec 5, Cor 5.9. The following $$L^1(\mu)$$ means completion of $$L_1$$ integrable step functions space completion under $$L_1$$ norm. And all maps in $$L^1(\mu)$$ are targeted to either reals. Actually, it works for even for all finite dimensional Banach spaces.

Cor 5.9 Let $$f$$ be $$\mu-$$measurable. Then $$f\in L^1(\mu)$$ iff $$|f|\in L^1(\mu)$$. More generally, if there exists an element $$g\in L^1(\mu)$$ s.t. $$g\geq 0$$ and $$|f|\leq g$$, then $$f\in L^1(\mu)$$.

$$\mu-$$measurable means that $$f$$ is pointwise limit of sequence of steps funcions upto difference over a set of measure $$0$$. The proof is basically done by DCT. In general, $$\mu-$$measurable is not necessarily in $$L^1$$ due to non-integrability.

$$\textbf{Q:}$$ The book says above corollary explains the role of positivity in integration theory. What is the role of positivity? Why it explains it? All I could see is the following. It basically says $$f\in L^1$$ iff $$f^+,f^-\in L^1$$.