I've been reading Paolini-Stepanov arcticle and in section 4, at page 6, they define a metric current from a transport: $$T_{\eta}(\omega)=\int_{\Theta}[[\theta]](\omega)d\eta(\theta),$$ which satisfies $$\mu_{T_\eta}\leq\int_{\Theta}\mu_{[[\theta]]}d\eta.$$ Where $\Theta$ is the set of Lipschitz curves, $\mu_T$ is the mass of the current and $[[\theta]]$ is the current associated to a Lipschitz curve.

Then they build an example in which this inequality is strict. In the example we take an $\eta_1$ concentrated over horizontal segments in $Q=[0,1]^2$ (which define the set $\Theta_1$) going from left to right and defined on a Borel set $e$ to be the 1-Length of the set of the starting points of the curves in $e$ $$\eta_1(e):=\mathcal{H}^1(e_0(e\cap\Theta_1).$$ Then they build $\eta_2$ in the same way on the vertical segments ion $Q$. Now they show that $T_{\eta_1+\eta_2}$ satisfies the inequality above in a strict sense.

The conclusion of the paper is that any acyclic normal current comes from a transport and have equality in that relation.

Since the obtained current cannot not be acyclic or normal, I think I am missing something. What am I missing?

Addendum: during the definition of this current they represent them with the notation $$T_{\eta_1}=\bar{e}_1\wedge\mathcal{L}^2\llcorner Q$$ Which is alien to me. Maybe by understanding it I could see the problem above.