= Solution
Write a <Blaschke product> as $B(z)=\lambda z^m\prod_n b_{a_n}(z)$, where $a_n\ne0$, $|\lambda|=1$, and the zeros are counted with <multiplicity>. With normalized angular measure $dm=d\theta/(2\pi)$, <Jensen's formula> for each <Blaschke factor> gives
$$
\int\log|b_a(re^{i\theta})|\,dm=\log\max(r,|a|).
$$
Indeed, the mean of $\log|a-re^{i\theta}|$ is $\log\max(r,|a|)$, while $\log|1-\overline a re^{i\theta}|$ has mean zero by the <mean value property for harmonic functions>. Each factor has modulus at most one. Applying the <monotone convergence theorem> to the negative logarithms of the finite products yields the <radial logarithmic mean of a Blaschke product>:
$$
\int\log|B(re^{i\theta})|\,dm=m\log r+\sum_n\log\max(r,|a_n|).
$$
The <Blaschke condition> implies $\sum_n-\log|a_n|<\infty$: only finitely many zeros have modulus below $1/2$, and $-\log t\le2(1-t)$ for $1/2\le t<1$. Every summand tends to zero and is bounded in absolute value by $-\log|a_n|$. The <dominated convergence theorem> for this series, together with $m\log r\to0$, therefore gives
$$
\boxed{\lim_{r\uparrow1}\int\log|B(re^{i\theta})|\,dm=0.}
$$
This also covers finite products and constant unimodular products. Zeros on a circle create integrable logarithmic singularities and do not change the conclusion.
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