Approximate versions of Cauchy's functional equation (Q1892743)

The authors prove that if \(f,a,b: \mathbb{R}\to \mathbb{R}\) are measurable functions and there is a \(J\in \mathbb{R}\) such that, for every \(\varepsilon> 0\), \(\mu(\{(x, y): | f(x+ y)- a(x)- b(y)- J|\geq \varepsilon\})\) is finite, then, for some \(\gamma\) and \(\beta\), \(f(x)= \gamma x+ \beta\) almost everywhere. It is also shown that if \(f\in L^ 1[0, \alpha]\) for all \(\alpha> 0\) and, for almost all \(x\), \(\lim_{u\to \infty} {1\over u} \int^ u_ 0 (f(x+ y)- f(x)- f(y)) dy= 0\), then for some \(\gamma\), \(f(x)= \gamma x\) for almost all \(x\geq 0\).