A Hamel coefficient has dense graph and a nonmeasurable kernel

F1 applies under A1. It gives additivity and rational linearity, $f(b)=1$, and $0\mathrm{\ne }w\in W$. In particular $b\mathrm{\ne }0$, $\mathbb{Q}w\subseteq W$, and $f^{-1}\{q\}=qb+W$: subtract qb and apply additivity in either direction. For any u<v choose by F2 a rational r strictly between $\min (u\mathrm{/}w,v\mathrm{/}w)$ and $\max (u\mathrm{/}w,v\mathrm{/}w)$. Then $rw\in (u,v)$, with the order reversed when w<0. Thus W is dense, and translation shows every fiber $qb+W$ is dense.

For any open rectangle $(u,v)\times (c,d)$, F2 gives rational q in (c,d); step 1.1 gives x in $(u,v)\cap (qb+W)$, so $(x,f(x))=(x,q)$ lies in the rectangle. Such rectangles form a basis, proving graph density. Taking (c,d) wholly above any prescribed M or wholly below -M shows both unboundedness assertions on each nondegenerate interval, whose interior contains an open interval. For any x_0 and $\delta >0$, density gives x with $\mathrm{\mid }x-x_0\mathrm{\mid }<\delta$ and $f(x)\in (f(x_0)+2,f(x_0)+3)$. This violates F8 with $ϵ=1$, so f is continuous at no x_0.

Suppose W measurable and put $W_m=W\cap [-m,m]$, m a positive integer. F5 and F6, licensed by A1, make these measurable with finite measure $a_m\le 2m$. Distinct rational translates $qb+W$ are disjoint: an equality $qb+w=rb+w^{\mathrm{\prime }}$ gives q=r after applying f and step 1.1. F2 enumerates the infinitely many rationals q with $\mathrm{\mid }qb\mathrm{\mid }<1$; infinitude follows from density in $(-1\mathrm{/}\mathrm{\mid }b\mathrm{\mid },1\mathrm{/}\mathrm{\mid }b\mathrm{\mid })$, since any finite list can be avoided in a smaller subinterval. The sets $W_m+qb$ along this enumeration are pairwise disjoint and all lie in $[-m-1,m+1]$. By F4 each has measure a_m. If $a_m>0$, choose by F3 a natural N with $N{a}_m>2m+2$. Finite additivity F5 and the enclosing interval value F6 then give $N{a}_m\le 2m+2$, contradiction. Hence every a_m is zero.

By F3, $W={\bigcup }_{m\ge 1}{W}_m$. Disjointizing this sequence and using F5's countable additivity shows W null. All cosets $qb+W$ are null by F4, and their countable union is $\mathbb{R}$ by F1's rational coefficient decomposition and F2's enumeration. Disjointization and F5 again make $\mathbb{R}$ null, contradicting F6's value one on [0,1] and monotonicity. Thus W is not measurable. Finally {0} is closed and Borel, so if f were Lebesgue measurable, F7 would make $f^{-1}\{0\}=W$ measurable, a contradiction. QED.