An isometry preserves the finite hyperbolic volume integral

Since $f$ is an isometry, $f^{\ast }{g}_N=g_M$. In any coordinate charts, if $J=Df$ is the coordinate derivative, then the metric matrices satisfy $G_M=J^{\mathsf{T}}(G_N\circ f)J$. Taking determinants gives $\sqrt{\det G_M}=\mathrm{\mid }\det J\mathrm{\mid }\sqrt{\det (G_N\circ f)}$. Thus $f$ pulls the Riemannian volume density of $N$ back to that of $M$, regardless of whether it preserves orientation.

Consider the set of ALL coordinate-neighborhood pairs $(U,V)$ with $f(U)\subset V$. It covers $M$. Fix a countable basis of $M$; for each basis member contained in some such $U$, use [A1] to select one containing pair. Their selected $U$ neighborhoods form a countable cover $(U_j{)}_{j\ge 1}$, with corresponding $V_j$. Thus no point-indexed choice of charts is required. Make it disjoint by setting $A_1=U_1$ and $A_j=U_j\setminus {\bigcup }_{i<j}{U}_i$; these are Borel, disjoint and cover $M$. In coordinates on $U_j,V_j$, apply [F2] to the nonnegative measurable function equal to the target density times the indicator of $f(A_j)$. The Jacobian identity in step 1.1 yields ${\mathrm{Vol}}_M(A_j)={\mathrm{Vol}}_N(f(A_j))$. The sets $f(A_j)$ are disjoint and cover $N$.

The indicators of the finite unions ${\bigcup }_{j\le n}{A}_j$ increase to $1_M$. By [F3] and [F4], the volume of $M$ is the limit of the sums ${\sum }_{j\le n}{\mathrm{Vol}}_M(A_j)$, and the same holds for $N$ and the sets $f(A_j)$. These finite sums agree term by term by step 2.1; their limits therefore agree in $[0,\mathrm{\infty }]$. If one volume is finite, their common value is finite.