The Gauss map preserves Gauss measure

By [F3], $\log 2>\log 1=0$, so $h$ is positive, bounded by $1\mathrm{/}\log 2$, and continuous on $[0,1]$. By [F4] it is Riemann integrable on every closed subinterval. The derivative of $\log (1+x)$ is $1\mathrm{/}(1+x)$ by [F2], using the translated difference quotient, also one-sided at subinterval endpoints. Thus [F5] and [F6] yield ${\int }_{[a,b]}hd\lambda =[\log (1+b)-\log (1+a)]\mathrm{/}\log 2$ for $0\le a<b\le 1$. Singletons and countable sets have zero density integral because they are null by [F7] and $h$ is bounded. Consequently endpoints do not change this interval value. By [F1] the density defines a Borel measure, and the value with $a=0,b=1$ is one; removing the endpoint 1 does not change it.

The Borel sets $J_n=(1\mathrm{/}(n+1),1\mathrm{/}n]\cap X$, $n\ge 1$, partition $(0,1)$. On $J_n$, $G(x)=1\mathrm{/}x-n$, and on the singleton $\{0\}$ it is zero. Each branch is the restriction of a continuous real function and takes values in $[0,1)$, so for every open subset of $X$ its inverse image is a countable union of Borel branch inverse images and possibly $\{0\}$. This proves Borel measurability. For $0<s<1$, the exact inverse image is $G^{-1}[0,s]=\{0\}\cup {\bigcup }_{n\ge 1}([1\mathrm{/}(n+s),1\mathrm{/}n]\cap X)$. The displayed intervals are pairwise disjoint because $1\mathrm{/}(n+s)>1\mathrm{/}(n+1)$. The only endpoint outside $X$ is 1 when $n=1$.

Using the interval integral of step 1.1 and countable additivity, $\mu (G^{-1}[0,s])$ is $1\mathrm{/}\log 2$ times the sum over $n\ge 1$ of $\log (1+1\mathrm{/}n)-\log (1+1\mathrm{/}(n+s))$. The quotient law [F3] and the identity $(1+1\mathrm{/}n)\mathrm{/}(1+1\mathrm{/}(n+s))=(1+s\mathrm{/}n)\mathrm{/}(1+s\mathrm{/}(n+1))$ rewrite the partial sum through $M$ as $\log (1+s)-\log (1+s\mathrm{/}(M+1))$. All original summands are nonnegative. Continuity of $\log$ at 1 gives the limit $\log (1+s)$, so $\mu (G^{-1}[0,s])=\log (1+s)\mathrm{/}\log 2=\mu ([0,s])$. For $s=0$, the preimage is $\{0\}\cup \{1\mathrm{/}n:n\ge 2\}$, an explicitly enumerated countable null set, and both masses are zero. The full space $X$ also has equal inverse-image mass one.

The family consisting of $X$, the empty set, and all $[0,s]$ with $0\le s<1$ is a pi-system. It generates the Borel sets of $X$: complements give $(s,1)$, and increasing unions of initial closed intervals give $[0,b)$; intersections give ordinary open intervals, which form a countable rational-endpoint base for the interval topology. Conversely all generators are Borel. The measure is finite and $G$ is measurable, so [F8] applies to step 2.1 and proves preservation for every Borel set. By [F9] the map is measurable and preserving on the completion as well. Countable choice is inherited in the Lebesgue/Riemann comparison, null-set and completion suppliers; the branch sums and partial-sum telescoping make no choices. The rational-base assertion uses [F10]. The rational-base assertion uses [F11].

The corresponding inverse-branch density calculation can also be seen directly. On $0<y<1$, the inverse branch is ${\eta }_n(y)=1\mathrm{/}(n+y)$, with $\mathrm{\mid }{\eta }_n^{\mathrm{\prime }}(y)\mathrm{\mid }=1\mathrm{/}(n+y{)}^2$. Hence $h({\eta }_n(y))\mathrm{\mid }{\eta }_n^{\mathrm{\prime }}(y)\mathrm{\mid }=1\mathrm{/}[\log 2(n+y)(n+y+1)]$. The algebraic identity $1\mathrm{/}[(n+y)(n+y+1)]=1\mathrm{/}(n+y)-1\mathrm{/}(n+y+1)$ gives the partial sum $[1\mathrm{/}(1+y)-1\mathrm{/}(M+1+y)]\mathrm{/}\log 2$, tending to $h(y)$. This verifies the density balance numerically; the interval proof in steps 1.1–3.1 already establishes measure preservation without assuming a change-of-variables theorem. The separate endpoint computations in step 2.1 account for $y=0$.