Let \(A\) be the weighted Bergman space of analytic functions in \(L^ 2(\Omega, \mu)\), where \(\Omega\) is an open set in \(\mathbb{C}^ n\), and \(\mu\) is a suitable measure on \(\Omega\). Let \(f\) be a measurable function on \(\Omega\). The big Hankel operator \(H_ f\) is defined for \(g\in A\) via \(H_ f(g)= (1-P)(fg)\), where \(P\) is the orthogonal projection from \(L^ 2(\Omega, \mu)\) onto \(A\). The author shows that various mapping properties of \(H_ f\) (boundedness, compactness, Schatten \(S_ p\)) are preserved when the measure \(\mu\) is multiplied by a positive function \(\varphi\) that, outside a compact subset of \(\Omega\), is bounded and bounded away from zero. However, the corresponding result is false for the small Hankel operator \(\widetilde {H}_ f\) defined by \(\widetilde {H}_ f(g)= \overline {P}(fg)\), where \(\overline {P}\) is the orthogonal projection of \(L^ 2(\Omega,\mu)\) onto the conjugate analytic functions.