Erdős problem 1126

If $$f(x+y)=f(x)+f(y)$$ for almost all $x,y\in \mathbb{R}$ then there exists a function $g$ such that $$g(x+y)=g(x)+g(y)$$ for all $x,y\in\mathbb{R}$ such that $f(x)=g(x)$ for almost all $x$.