Theorem

Let $u,v:X\to \mathbb{R}$ be measurable functions where $(X,\mathcal{M}),(\mathbb{R},\mathcal{B}(\mathbb{R}))$ are measurable spaces then: 1. $u+v,\ uv,\ |u|:X\to \mathbb{R}$ and $u+iv:X\to \mathbb{C}$ are measurable. 2. If $f,g:X\to \mathbb{C}$ are measurable, then $$f+g,\ fg,\ Re(f),\ Im(f):X\to \mathbb{C}$$are measurable. 3. If $E\subseteq X$ then $$E\in\mathcal{M}\iff \mathbb{1}_{E}\text{ is measurable}$$ 4. If $f:X\to \mathbb{C}$ is measurable, we can write $f=\alpha \cdot|f|$ where $\alpha$ and $|f|$ are measurable and $|\alpha|=1$