S. Nakanishi gave some interesting results on the  weak convergence 
of measures on the union $X$ of metric spaces 
$(X_{\alpha}, d_{\alpha}) \ (\alpha \in \Sigma),$ 
endowed with the finest topology for which all the canonical 
injections of $X_{\alpha}$ to $X$ are continiuous. Our aim in this
note is to extend her results to the case where each component space 
$X_{\alpha}$ is simply a Hausdorff space.