I am studying the paper Blumenthal - Statistical properties for compositions of standard maps with increasing coefficent. I have a problem to understand how the distortion estimates are used. The setting is the following: I have a family $\{ F_N\}$ of diffeomorphisms of the torus $\mathbb{T}^2$. Every map is associated to a number $L_N \to \infty$ as $N \to \infty$. I define the family, for $K>M$ $$F_M^K:= F_K \circ \dotsb\circ F_{M+1} \circ F_M. $$ I take a curve $\gamma \in \mathbb{T}^2$ of the form $(x,g(x))$, $g \colon I_g \subset [0,1) \to [0,1) $. Now consider $J \subset I_g$ and set $$ \tilde{\gamma}:= (F_M^{K})(\gamma\rvert_J). $$ Briefly speaking, but it is not relevant to the question, in the paper the geometry of the map is such that I can still write this curve $\tilde{\gamma}$ as of the form $(x,h(x))$.
In Lemma 2.7(c) they prove that, if $p,q \in (F^K_M)^{-1}\tilde{\gamma}$ then $$\frac{\lVert(dF_M^K)_p\rvert_{T\gamma}\rVert}{\lVert(dF_M^K)_q\rvert_{T\gamma}\rVert}= 1 + O(L_M).$$ This is called distortion estimate. What I do not understand, is how do they use this estimate in the rest of the paper? In proposition 2.8, they claim $$\int_{\tilde{\gamma}}\frac{d\mathrm{Leb}_{\tilde{\gamma}}}{\lVert dF_M^{K} \rVert \circ F_M^{K}} X_{R_{i,j}}\circ F_M = (1+O(L_m)) \mathrm{Leb}_{\gamma}((F_M^K)^{-1}\tilde{\gamma})\int_{\tilde{\gamma}} X_{R_{i,j}}\circ F_M$$ and they claim this identity follows directly from the estimate in lemma 2.7(c). (What is $R_{i,j}$ is irrelevant to my question.)
My questions are the following:
(1) How do they derive this identity using the distortion estimate?
(2) Can anyone provide a formal definition for the "Lebesgue measure" in a curve $\mathrm{Leb}_{\gamma}$ as used in the paper?
(3) What is the relation between this definition of Lebesgue measure on the curve and the distortion estimates?
