Suppose $u$ is harmonic on the $B(0,r)$ and continuous on $\overline {B(0,r)}$ then $$u(0) = \frac {1} {2 \pi} \int_{0}^{2 \pi} u \left (re^{i \theta} \right )\ d\theta.$$
I know the result for any $0 \lt r_1 \lt r.$ Can it be extended for $r$ as well by taking limits as $r_1 \to r^{-}\ $? I think we can do that using DCT. Let $M = \|u\|_{\infty}.$ Take a sequence $\{r_n\}_{n \geq 1}$ of positive reals such that $r_n \big\uparrow\ r.$ Define $u_n \left ( e^{i \theta} \right ) = u \left (r_n e^{i \theta} \right ).$ Then $u_n \left (e^{i \theta} \right ) \to u \left (r e^{i \theta} \right )$ for all $\theta$ by the continuity of $u$ on $\partial B (0,r)$ and $|u_n| \leq M.$ Since we are in a finite measure space constant functions are integrable and hence we can apply DCT to conclude the desired equality. Am I right?