In fact a weaker version of Aleksandrov's theorem is true for subharmonic functions. Since $\Delta u$ is a Radon measure, the following result follows from Proposition 4.4 in [1].

>**Theorem.** If $u:\Omega\to\mathbb{R}$ is subharmonic and locally integrable, then
for almost all $x\in\Omega$ there exist a vector $Du(x)$ and a matrix $D^2u(x)$ such that
for all $1\leq p<n/(n-2)$ ($1\leq p<\infty$, if $n=2$)
$$
\left(\frac{1}{|B(x,r)|}\int_{B(x,r)}|u(y)-u(x)-u(x)(y-x)-\frac{1}{2}(y-x)^TD^2u(x)(y-x)|^p\right)^{1/p}\\=o(r^2).
$$

I believe, the matrix $D^2u(x)$ is symmetric, but it is not explicitly stated in [AG].

As a corollary, the authors prove Imomkulov's theorem (see the answer of user111), but they do it independently (21 years later!) since they do not quote the work of Imomkulov.

If you want to compare it with the statement of Aleksandrov's theorem, see https://mathoverflow.net/q/352421/121665.

**[AG] G. Alberti, S. Bianchini, C. G. Stefano,** On the $L^p$-differentiability of certain classes of functions. *Rev. Mat. Iberoam.* 30 (2014), no. 1, 349–367.