Theorem (Closure of Measurability)

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 $|\alpha|=1$

Theorem (Measurability of Continuous Functions)

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function, then automatically $$f:(\mathbb{R},\mathcal{B}(\mathbb{R}))\to(\mathbb{R},\mathcal{B}(\mathbb{R}))$$i.e. $f$ is a measurable function when mapping between Borel σ-algebra spaces.

Theorem (1.8)

Let $(X,\mathscr{M})$ be a Measure Space and let $u,v:X\to \mathbb{R}$ be two Measurable Functions where $\mathbb{R}$ is equipped with the Standard topology. Suppose that $(Y,\mathscr{T})$ is a Topological Space and $\Phi:\mathbb{R}^2\to Y$ is Continuous (here $\mathbb{R}^{2}$ is also equipped with standard topology) then $h:X\to Y$ defined as $$h(x)=\Phi(u(x),v(x))$$ is measurable.