I am trying to find any techniques to analyze the measure of an image of a set under an argmax function.
For example, let $\Omega\subset\mathbb{R}^n$ be compact and $\phi:\Omega\to\mathbb{R}$ be Lipschitz. Consider a maximization map $M:B_1(0)\to\mathbb{R}$ defined by $$M(\xi) := \max\limits_{x\in\Omega}\left[\phi(x)-\frac{1}{2}|x-\xi|^2\right].$$
It is my understanding that we can find a measurable selection from argmax. That is, there exists a measurable map $X^\star : B_1(0) \to \Omega$ with the property $$M(\xi) = \phi(X^\star(\xi))-\frac{1}{2}|X^\star(\xi)-\xi|^2$$ for all $\xi\in B_1(0)$.
I am interested in finding techniques for computing (or bounding) the measure of the image $X^\star(B_r(0))$ for $r>0$. Are there any standard techniques which are useful when dealing with argmax?
I have not dealt with these measurable selection theorems before. I guess I also am interested in how I might prove this image is measurable to begin with...
In particular, I would be happy to find conditions such that for all $r>0$, the Lebesgue measure of $X^\star(B_r(0))$ is strictly positive. Does anyone have ideas how I might be able to attack this?
