potential $\psi$, the condition $\theta+dd^c\varphi_j\geq\gamma$ implies that $\psi+\varphi_j-q$ is pluri-subharmonic. The upper enveloppe $\psi+\varphi-q$ is thus also pluri-subharmonic, which means that $T=\theta+dd^c\varphi\geq dd^cq$; letting $dd^cq$ tend to $\gamma$, we get $T\geq\gamma$, as desired.\
Since any two closed almost positive currents with equivalent singularities have the same Lelong numbers, the Lelong numbers of $\inf T_j$ do not depend on the specific choice of the current. In fact, it is immediate to check from the definitions that $$\nu(\inf_{j\in J} T_j,x)=\inf_{j\in J}\nu(T_j,x).$$\
As a particular case of the above construction, there exists a closed almost positive $(1,1)$-current $T_{\min,\gamma}\in\alpha[\gamma]$ which is a least element in $(\alpha[\gamma],\preceq)$. $T_{\min,\gamma}$ is well defined modulo $dd^cL^{\infty}$, and we call it a current with minimal singularities in $\alpha$, for the given lower bound $\gamma$. When $\gamma=0$ and $\alpha$ is pseudo-effective, we just write $T_{\min}=T_{\min,0}$, and call it a positive current with minimal singularities in $\alpha$. It must be noticed that, even for a big class $\alpha$, $T_{\min}$ will be a Kähler current only in the trivial case:

A pseudo-effective class $\alpha$ contains a positive current with minimal singularities $T_{\min}$ which is a Kähler current iff $\alpha$ is a Kähler class.

$Proof$: we can write $T_{\min}=\theta+dd^c\varphi$ with $\theta$ a smooth 