I am a little confused about Corollary 4.13 in paper of Leininger and Aramayona. Let's keep everything simple that we denote by $S,\tau(S),\mathcal{S}(S),ML(S)$ closed orientable surface of genus $g\geq 2$, Teichmuller space of $S$, the set of simple closed curves up to isotopy, and the space of measured laminations of $S$ respectively. The statement is the following:
Corollary 4.13: The length function $\ell:\tau(S)\times\mathcal{S}(S)\to\mathbb{R}$ extends to a continuous function $\ell':\tau(S)\times ML(S)\to\mathbb{R}$ which is homogeneous in the second factor.
I am confused about what $\ell'$ looks like. Take any $(X,f)\in\tau(S)$, $\alpha\in\mathcal{S}(S)$, and a measured lamination $(\mathscr{L},\lambda)\in ML(S)$. We embed Teichmuller space, measured laminations into the space of geodesic currents where the length function $\ell_{\alpha}(X)=i(L_X,\alpha)$ where $L_X$ denotes the Liouville current, $\alpha$ a current represented by the closed geodesic and $i$ denotes the intersection form of geodesic currents.
Now, can I interpret $\ell'_{\mathscr{L}}(X)$ as $i(L_X,\lambda)$? However, I have trouble understanding $i(L_X,\lambda)$. Here is my thought so far: the support of the measure $L_X\times\lambda$ lies in $\mathscr{L}\times E$ where $E$ denotes the set of geodesics intersects the lamination $\mathscr{L}$ transversely. But I have trouble proceeding.
Thanks for your help!
In brief, although your description of the support does not really make sense globally, with a slight reformulation one can make sense of it locally. One checks that these local descriptions fit together to give a global measure on $S$ that is supported on $\lambda$. Integrating this measure defines $i(L_X,\lambda)$ (this explanation was given by Thurston in early lectures on the topic).
Here are more details. Around each point of $\lambda$ there exists a lamination coordinate chart of the form $$f : (a,b) \times (0,1) \approx U \subset S $$ together with a compact (and totally disconnected) subset $E \subset (0,1)$ such that $f\bigl( (a,b) \times E \bigr) = U \cap \lambda$, and such that for each $t \in E$ the restriction $f \mid (a,b) \times \{t\}$ is an isometry onto a leaf segment of $\lambda$. One obtains a measure on $U$ that is supported on $U \cap \lambda$, which is expressed as a Fubini product of the Lebesgue measure on $(a,b)$ times the transverse measure on $E$. Where two lamination coordinate charts overlap, the two measures agree. These measures therefore fit together to define a global measure on $\Lambda$. Integrating that global measure yields the number $i(L_X,\lambda)$.
Let me add that the special case where $\lambda$ is a simple closed curve is incorporated into the general description above: $E$ is just a single point with an atomic mass of some value $\delta > 0$, equal to the transverse measure on the simple closed curve $\lambda$, and so in that case $i(L_x,\lambda)$ is just $\text{Length}(\lambda) \times \delta$.