If \(f(x)+g(x)>a\), there is a rational \(q\) with \(f(x)>q\) and \(g(x)>a-q\), so \(\{f+g>a\}\) is a countable union of intersections of measurable sets. Neither the union of \(\{f>a\}\) and \(\{g>a\}\) nor continuity or boundedness justifies measurability.
A function f: X → ℝ is measurable with respect to a σ-algebra \(\mathcal{A}\) on X if the preimage \(f^{-1}(B)\) belongs to \(\mathcal{A}\) for every Borel set B of ℝ.