Proposition

Let $(X,\mathscr{M})$ be a measurable space, let $f:X\to \mathbb{R}$ then $$f\text{ is measurable}\iff\begin{cases} f^{-1}((-\infty,a))\in\mathscr{M} \\ \text{or}\\ f^{-1}((a,\infty))\in\mathscr{M} \end{cases}\quad \text{for }a\in\mathbb{R}$$and if $f:X\to\overline{\mathbb{R}}$ then $$f\text{ measurable}\iff f^{-1}((a,+\infty])\in\mathscr{M}$$ ## Remark What this proposition gets to the heart of is that given our original definition of measurability where we require the preimage of all Open sets to be in our σ-algebra, we show here that any open set is just the union of open intervals. Though this fact we show that the preimage of this specific open interval implies all other open intervals satisfy the same condition and hence any open set does too.