I am trying to understand why the following defines a measure. Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function.
Define a measure $\mu$ on $\mathcal{B}(\mathbb{R}^d)$ by $$\mu(E) = \mathcal{L}^d(\partial \phi (E))$$ where $\mathcal{L}^d$ is the Lebesgue measure in $\mathbb{R}^d.$
To see that by this definition we get a measure (and that the definition is well posed) we need to check that the image $\partial \phi (E)$ is a Lebesgue-measurable subset of $\mathbb{R}^d,$ and that $\mu$ is sigma-additive.
I have come to no conclusion while trying to show that the image is measurable.
In one dimension sigma-additivity follows from monotonicity of the subgradient, so I would expect to use monotonicity also in higher dimensions, but I don´t see how.