Differentiation of L-one functions for a doubling weight

First suppose $h\in L^1(v)$ globally and put $\mu =vd\lambda$. For $g\in C_c$ and $G=\mathrm{\mid }h-g\mathrm{\mid }$, the centred weighted oscillation $Dh(x):={\mathrm{lim\hspace{0.17em}sup}}_{r\downarrow 0}\mu (B(x,r){)}^{-1}{\int }_{B(x,r)}\mathrm{\mid }h(y)-h(x)\mathrm{\mid }d\mu (y)$ is at most $M^{v}G(x)+G(x)$, since the corresponding oscillation of continuous $g$ tends to zero. These limsups are measurable: for each fixed $x$ the integrals are continuous in positive radius by dominated convergence, hence rational radii suffice; joint measurability follows after choosing Borel representatives, and changing them on a null set does not change the a.e. assertion.

For $\eta >0$, $\{Dh>\eta \}\subseteq \{M^{v}G>\eta \mathrm{/}2\}\cup \{G>\eta \mathrm{/}2\}$. The weak bound and Chebyshev give $\mu (\{Dh>\eta \})\le 2(C(n,c_v)+1){\eta }^{-1}\mathrm{\parallel }h-g{\mathrm{\parallel }}_{L^1(v)}$. Taking the infimum over $g\in C_c$ using [F2] makes this measure zero. The countable union over $\eta =1\mathrm{/}k$ is null, so the centred absolute oscillation tends to zero a.e. For local $h$, apply this result to $h_m=h{\mathbf{1}}_{B(0,m+1)}\in L^1(v)$; on $B(0,m)$, sufficiently small centred balls see $h=h_m$. The countable union of these exceptional sets is null and ${\bigcup }_{m}B(0,m)={\mathbb{R}}^n$, proving the same conclusion for every local input.

Outside the null set of step 2.1, the absolute oscillation over any shrinking set $E_r\subseteq B(x,{\rho }_r)$ with ${\rho }_r\to 0$ and $\mu (B(x,{\rho }_r))\le C_x\mu (E_r)$ is bounded by $C_x$ times the centred oscillation. It therefore tends to zero, and the modulus of the difference between the average of $h$ and $h(x)$ is at most this oscillation. Balls and cubes shrinking nicely with the stipulated weighted comparability meet these hypotheses. This proves all the stated limits, on a common full-measure set.