  We use an observation of Bohr connecting Dirichlet series in the right half
plane $\mathbb{C}_+$ to power series on the polydisk to interpret Carlson's
theorem about integrals in the mean as a special case of the ergodic theorem by
considering any vertical line in the half plane as an ergodic flow on the
polytorus. Of particular interest is the imaginary axis because Carlson's
theorem for Lebesgue measure does not hold there. In this note, we construct
measures for which Carlson's theorem does hold on the imaginary axis for
functions in the Dirichlet series analog of the disk algebra
$\mathcal{A}(\mathbb{C}_+)$.
