  We consider among the Irwin series (1916) those which are the sums of the
reciprocals of the integers having $k$ occurrences of a given digit $d$ (in an
arbitrary base $b$). We formulate them as log-like integrals against suitable
measures on the unit interval. This leads to representations as geometrically
convergent alternating series. The alternating series involve the moments of
these measures. The measures converge for $k\to\infty$ to the Lebesgue measure
(up to a factor $b$). This puts Farhi theorem (2008) about $b\log b$ being the
limit of the Irwin sums for $k\to\infty$ in a new light. The moments obey
linear recurrences. This allows evaluating them numerically hence the Irwin
sums in a straightforward manner.
