Suppose we have a function $f \colon \mathbb{R}^n \to [0,\infty)$ that is continuous, decreasing, assumes a maximum at the origin, and Lebesgue integrable. Using the layer-cake formula one may write $f(x) = \int_0^{\infty} 1_{ \{f \geq s\}}$. What conditions must be put on a density $\phi$ for the following to happen? Is translations invariance a necessaity?
$$ \max_{w \in \mathbb{R}^n} \left\{\int_{0}^{\infty} \int_{\{f \geq s\} + w} \phi(x) dx ds\right\} = \int_0^{\infty} \max_{w \in \mathbb{R}^n} \left\{ \int_{\{f \geq s\} + w} \phi(x) dx \right\} ds? $$