Let $(E,\mathcal E,\lambda)$ be a measure space, $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$. It's easy to see that $$L^2(\mu)\ni w\mapsto\int|w|^2\:{\rm d}\mu=\left\|w\right\|_{L^2(\mu)}^2\tag1$$ is Fréchet differentiable with Fréchet derivative $2\langle w,\;\cdot\;\rangle_{L^2(\mu)}$ at $w\in L^2(\mu)$, since this is true for the squared norm of any $\mathbb R$-Hilbert space and can be easily proven by the Cauchy-Schwarz inequality.
Now assume that $f:E\to[0,\infty)$ is $\mathcal E$-measurable with $\{p=0\}\subseteq\{f=0\}$. Can we show that $$F(w):=\int|w|^2\:{\rm d}(f\lambda)\;\;\;\text{for }w\in L^2(\mu)$$ is Fréchet differentiable as well?