In the definition of subharmonic, the continuity assumption...

In the definition of subharmonic, the continuity assumption can be replaced by upper semi-continuity. Let $\varphi_z$ be as in Exercise 7.13. Suppose $v$ is upper semi-continuous on a region $\Omega$ and suppose $v$ satisfies the mean-value inequality on sufficiently small disks contained in $\Omega$. Define $$ v_\delta(z)=\int_{\mathbb{C}} v(w) \varphi_s(z-w) d A(w), $$ for $z \in \Omega$ and $\delta<\operatorname{dist}(z, \partial \Omega)$. (a) Prove that $v_z$ is subharmonic and decreases to $v$ (eventually) on compact subsets of $\Omega$. (d) Use part (a) or check the proofs directly using the definition of upper semicontinuity to show that the results of this chapter remain valid for this weaker definition of subharmonic.

In the definition of subharmonic, the continuity assumption can be replaced by upper semi-continuity. Let $\varphi_z$ be as in Exercise 7.13. Suppose $v$ is upper semi-continuous on a region $\Omega$ and suppose $v$ satisfies the mean-value inequality on sufficiently small disks contained in $\Omega$. Define
$$
v_\delta(z)=\int_{\mathbb{C}} v(w) \varphi_s(z-w) d A(w),
$$
for $z \in \Omega$ and $\delta<\operatorname{dist}(z, \partial \Omega)$.
(a) Prove that $v_z$ is subharmonic and decreases to $v$ (eventually) on compact subsets of $\Omega$.
(d) Use part (a) or check the proofs directly using the definition of upper semicontinuity to show that the results of this chapter remain valid for this weaker definition of subharmonic.

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