Conditions that guarantee the existence of a largest piece

Question

Let $m$ be the area measure on $R^2$.

Let $S$ be a nonempty set of measureable subsets of $R^2$ ("pieces").

Define the largest piece in $S$ as:

$$\arg \max_{s\in S} m(s)$$

I am looking for conditions on $S$ that guarantee that it has a largest piece

(i.e. necessary and sufficienet conditions guaranteeing that the maximum is attained).

I thought of using the extreme value theorem, which says that a continuous function on a compact set has a maximum. To use this theorem, I need a topology on $P(R^2)$ such that:

• The area measure $m$ is a continuous function from $S$ to $R$.
• It is easy to define necessary and sufficient conditions for the compactness of $S$.

Is there any standard topology on $P(R^2)$ with these properties?

What are the conditions on $S$?

Answer 1

One standard way of defining a metric on measurable subsets of $\mathbb{R}^2$ is the following: $d(A, B) = m(A \Delta B)$ where $A \Delta B$ is the symmetric difference of $A$ and $B$. If you identify sets which differ by measure zero sets then this defines a complete separable metric so compactness would mean closed and totally bounded. $S$ being closed is relatively easy to visualize but totally bounded is a little more technical.

Answer 2

The Hausdorff metric on compact subsets of a compact space makes the set of such subsets into a compact metric space. Lebesgue measure is not continuous on this space, but it is upper semicontinuous, i.e. if $C_n \to C$ in the Hausdorff metric, then $m(C) \ge \limsup_n m(C_n)$, and this is enough to ensure the existence of a "largest piece".