Let $f \in L_{1}^{w}[-1, 1]$, let $r_{n, m}(f)$ be a best rational $L_{1}^{w}$-approximation for $f$ with respect to real
rational functions of degree at most n in the numerator and of degree at most m in the denominator,
let $m = m(n)$, and let $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. Then we show that the counting measures of certain subsets of sign
changes of $f - r_{n,m}(f)$ converge weakly to the equilibrium measure on $[-1, 1]$ as $n\rightarrow \infty$.
Moreover, we prove discrepancy estimates between these counting measures and the equilibrium measure.
