Let $I$ be a finite nonempty set, $p,q_i$ be probability densities on a measurable space $(E,\mathcal E,\lambda)$ for $i\in I$, $w_i:E\to[0,1]$ be measurable with $\{q_i=0\}\subseteq\{w_ip=0\}$ for $i\in I$ with $\{p>0\}\subseteq\{\sum_iw_i=1\}$ and $g\in\mathcal L^1(\lambda)$ with $\{p=0\}\subseteq\{g=0\}$.
How can we minimize $$\sum_{i\in I}\frac1{n_i}\int_{\{\:q_i\:>\:0\:\}}\frac{|w_ig|^2}{q_i}\:{\rm d}\lambda\tag1$$ with respect to $(w_i)_{i\in I}$?
If $p,q_i$ would be (strictly) positive for all $i\in I$, $(1)$ could be written as $$\int\sum_{i\in I}\frac1{n_i}\frac{|w_ig|^2}{q_i}\:{\rm d}\lambda\tag2$$ and it is obvious that it's sufficient to perform a pointwise minimization of the integrand in $(2)$. Moreover, since $g$ is a constant in this context, we are then left with the following problem: Given $(q_i)_{i\in I}\subseteq[0,\infty)$, minimize $$\sum_{i\in I}\frac1{n_i}\frac{w_i^2}{q_i}\tag3$$ subject to $$\sum_{i\in I}w_i=1\tag4$$ with respect to $(w_i)_{i\in I}\subseteq[0,\infty)$. This problem is easily solved by the Lagrange multiplier theorem yielding the minimizer $$w_i=\frac{n_iq_i}{\sum_{j=1}^kn_jq_j}\tag5.$$
Can we apply the same argumentation to the original problem?