I need to show that an increasing function $f:\mathbb{R}\rightarrow \mathbb{R}$ is (borel)-mesurable
proof From the lecture we know that since $B(\mathbb{R})=\langle Q(\mathbb{R})^\sigma\rangle$, it's enough to show that $\forall E\in Q(\mathbb{R})\,\,\,f^{-1}(E)\in B(\mathbb{R})$. Therefore let $E$ be as above, then $E=\bigcup_{i=1}^n[a_i,b_i),\,\,\,s.t.a_1<b_1<a_2<b_2...$ then $$f^{-1}(E)=\bigcup_{i=1}^nf^{-1}([a_i,b_i))$$ Now we only need to show that $O_i=f^{-1}([a_i,b_i))$ are in $Q(\mathbb{R})$, and then by the characterisation of $B(\mathbb{R})$ we can conclude that $\bigcup_{i=1}^n O_i\in B(\mathbb{R})$.
To do so I only need to prove that the preimage of an intervall is again an intervall.
Thank you very much.