Let $(E,\mathcal E,\lambda)$ be a measure space, $f:E\to[0,\infty)$ be $\mathcal E$-measurable, $\mu:=f\lambda$ and $$E(p):=\int_{\{\:p\:>\:0\:\}}\frac1p\:{\rm d}\mu\in[0,\infty]$$ for $\mathcal E$-measurable $p:E\to[0,\infty)$.
I want to minimize $E(p)$ subject to $$\int p\:{\rm d}\lambda=1$$ using the method of Lagrange multipliers. (I already know the solution. The question really is how we can find it using Lagrange multipliers.)
Unfortunately I'm already failing to come up with a suitable Banach space in which we search the solution. (The main problem being that we can't simply choose an $L^q$-space and $E(p)$ might be $\infty$. I'm not sure, but since I'm willing to assume that there exists at least one $p$ such that $E(p)<\infty$, we might resolve the latter issue somehow.)