= Solution
Take velocities relative to a comoving observer at $O$. Spatial homogeneity means that changing the observer to $A$ gives the same relative velocity law, so
$$
v(r_B-r_A)=v(r_B-r_O)-v(r_A-r_O).
$$
With $O$ as origin this implies $v(0)=0$ and $v(a+b)=v(a)+v(b)$. A physical <velocity field> is continuous; additivity first gives rational homogeneity and then, by continuity, real homogeneity. Hence
$$
\boxed{v_i(r)=\sum_jH_{ij}r_j,\qquad H_{ij}=[v(e_j)]_i,}
$$
with coefficients independent of position. Without a regularity assumption, the additive functional equation alone admits non-linear pathological solutions; continuity is the usual implicit velocity-field assumption.
Isotropy requires $v(Rr)=Rv(r)$ for every spatial rotation, or $HR=RH$. Rotations about $e_i$ force $He_i$ to be parallel to $e_i$, and rotations exchanging coordinate directions force those diagonal coefficients to agree. Therefore $H_{ij}=H\delta_{ij}$ and $\boxed{v=Hr}$, the <Hubble law>.
Allow $H=H(t)$. Solving $\dot r=H(t)r$ gives $r(t)=a(t)x$ for a constant <comoving coordinate> $x$, where $a(t)=a(t_0)\exp(\int_{t_0}^tH(s)ds)$. Thus $\boxed{H=\dot a/a}$, with $a$ the <scale factor>.
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