If we use rescale the metrics further around the point $v=0$ such that the sequence of spaces collapse to the complete real line $\mathbb{R}$, then $d\underline{\nu}_{\infty}$ coincides with the standard Lebesgue measure. In particular, the singularity at $v=0$ disappears. This fact can be quickly seen by scaling-up the coordinates $v$. This is compatible with the general theory of Ricci-limit spaces. That is, due to Cheeger-Colding, the renormalized limit measure always splits off the Lebesgue measure of $\mathbb{R}$ if the limit space isometrically splits off $\mathbb{R}$ (see proposition 1.35 in [CC97] for more details).