Theorem (1.39)

Let $(X,\mathscr{M},\mu)$ be a Measure Space.

1. Suppose $f:X\to[0,+\infty]$ is a Measurable Function and $E\in\mathscr{M}$ is s.t. $$\int\limits _{E}f \, d\mu=0$$then $f=0$ a.e. on $E$
2. If $f$ is Integrable and $$\int\limits _{E}f \, d\mu =0\quad \forall E\in\mathscr{M}$$then $f=0$ a.e.
3. If $f$ is integrable and $$\left|\int\limits f \, d\mu \right|=\int\limits |f| \, d\mu$$then $\exists\theta \in[0,2\pi)$ s.t. $e^{i\theta}f=|f|$ a.e.