  We improve the discretization technique for weighted Lorentz norms by
eliminating all "non-degeneracy" restrictions on the involved weights. We use
the new method to provide equivalent estimates on the optimal constant $C$ such
that the inequality $$\left( \int_0^L (f^*(t))^{p_2} w(t)\,\mathrm{d}t
\right)^\frac 1{p_2} \le C \left( \int_0^L \left( \int_0^t u(s)\,\mathrm{d}s
\right)^{-\frac {p_1}\alpha} \left( \int_0^t (f^*(s))^\alpha u(s) \,\mathrm{d}s
\right)^\frac {p_1}\alpha v(t) \,\mathrm{d}t \right)^\frac 1{p_1}$$ holds for
all relevant measurable functions, where $L\in(0,\infty]$, $\alpha, p_1, p_2
\in (0,\infty)$ and $u$, $v$, $w$ are locally integrable weights, $u$ being
strictly positive. It the case of weights that would be otherwise excluded by
the restrictions, it is shown that additional limit terms naturally appear in
the characterizations of the optimal $C$. A weak analogue for $p_1=\infty$ is
also presented.
