By Lemma 2.1 the volume mass is uniformly upper bounded. By Federer-Fleming compactness, subsequentially $L_{i}\to L$ in the flat topology for some Lagrangian integral current $L$ homologous to $L_{0}$. This implies $\int_{L_{k}}g\text{Re}\Omega\to\int_{L}g\text{Re}\Omega$ for any $C^{\infty}$ test function $g$, even though $\liminf_{i}Mass(L_{i})$ may be strictly greater than $Mass(L)$, as we do not assume varifold convergence.

We focus on a coordinate ball. The $n$-currents $f_{L_{k}}L_{k}$ can be viewed as a collection of ${2n\choose n}$ signed measures $g\mapsto\int_{L_{k}}f_{L_{k}}gdx_{i_{1}}\wedge\ldots dx_{i_{n}}$. Each of these measures are bounded by the measure

$$g\mapsto(\sup\left\lVert f_{L_{i}}\right\rVert_{L^{\infty}})\frac{C}{\sin\epsilon}\int_{L_{k}}g\text{Re}\Omega,$$

whose total mass is uniformly bounded for all $k$. By the weak compactness of measures, subsequentially these signed measures converge, and the limiting signed measures have $L^{\infty}$ Radon-Nykodim derivatives $a_{i_{1}\ldots i_{n}}(x)$ with respect to the measure $g\mapsto\int_{L}g\text{Re}\Omega$:

$$\int_{L_{k}}f_{L_{k}}gdx_{i_{1}}\wedge\ldots dx_{i_{n}}\to\int_{L}ga_{i_{1}\ldots i_{n}}\text{Re}\Omega.$$

Thus inside the coordinate ball, the currents $f_{L_{k}}L_{k}$ converge to $\lim_{k}f_{L_{k}}L_{k}$:

$$\eta\mapsto\int_{L}\sum_{i_{1}<i_{2}\ldots<i_{n}}\eta(\partial_{i_{1}}\wedge\ldots\partial_{i_{n}})a_{i_{1}\ldots i_{n}}\text{Re}\Omega,$$

where $\eta$ is any test $n$-form.

Now $L$ is an integral current, so $\mathcal{H}^{n}$-a.e. $y\in\text{supp}(L)$ there is a well defined tangent space $T_{y}L$ and a local integer multiplicity $\Theta(y)$. Recall a blow up limit of an $n$-current $N$ at a point $y\in X$ refers to a subsequential limit of the currents on $T_{y}X$ as $r\to 0$:

$$\eta\mapsto\int_{N}\text{rescale}_{y,r}^{*}\eta,\quad\text{rescale}_{y,r}:x\mapsto\frac{x}{r}\text{ in the geodesic coordinates around $y$}.$$

For a.e $y\in\text{supp}(L)$, there is a unique blow up limit for the current $\lim_{k}f_{L_{k}}L_{k}$, which is

$$\eta\mapsto\int_{T_{y}L}\sum\eta(\partial_{i_{1}}\wedge\ldots\partial_{i_{n}})a_{i_{1}\ldots i_{n}}(y)\Theta(y)\text{Re}\Omega,$$

whose ${2n\choose n}$ component signed measures are just constant multiples of the Lebesgue measure on $T_{y}X$.

Observe that the weak formulation (54) passes to the limit:

$$\int_{L}\lambda\wedge\chi=-(\lim_{k}f_{L_{k}}L_{k})(d\chi).$$

Thus the blow up limit of $\lim_{k}f_{L_{k}}L_{k}$ at a.e. $y\in\text{supp}(L)$ is in fact a closed current. Consequently, the polyvector

$$\sum_{i_{1}<i_{2}\ldots<i_{n}}a_{i_{1}\ldots i_{n}}\partial_{i_{1}}\wedge\ldots\partial_{i_{n}}$$

must be a pure tensor lying in $\Lambda^{n}T_{y}L\subset\Lambda^{n}T_{y}X$. Hence

$$\lim_{k}f_{L_{k}}L_{k}=f_{L}L$$

for some $L^{\infty}$-function $f_{L}$. ∎