Gauss measure ($d\mu =dx/[(\log 2)(1+x)]$)

The invariant measure for the Gauss continued-fraction map with density $1/[(\log 2)(1+x)]$ on $[0,1]$. Inverse branches $x=1/(n+y)$ give transformed density $(\log 2{)}^{-1}{\sum }_{n\ge 1}[(n+y{)}^{-1}-(n+y+1{)}^{-1}]$, which telescopes to the same density.