= Absolutely continuous representative along free characteristics
{title2=$F(t)=C+\int_0^tG(s)\,ds$}
A <distributional weak solution> of $\partial_tf+v\cdot\nabla_xf=g$, with locally integrable $f,g$, becomes $\partial_tF=G$ after the shear $F(t,x,v)=f(t,x+tv,v)$. For almost every characteristic label, there is an <absolutely continuous function> representing $F$, and its increments are the <integrals> of $G$ for every pair of times. Arbitrary modifications on time slices can destroy this every-time identity without changing the distributional solution.
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