How to construct a Bernstein set and what are their applications?

Solution 1:

One uses the axiom of choice in some form to construct a Bernstein set. The most straightforward way is by transfinite recursion. There are $2^\omega$ uncountable closed subsets of $\Bbb R$, so we can list them as $\{F_\xi:\xi<2^\omega\}$, and it can be proved that each of them has cardinality $2^\omega$.

Now suppose that $\eta<2^\omega$, and for each $\xi<\eta$ you’ve chosen points $x_\xi,y_\xi\in F_\xi$ so that all of these points are distinct. Let $X_\eta=\{x_\xi:\xi<\eta\}$ and $Y_\eta=\{y_\xi:\xi<\eta\}$. Then $|X_\eta\cup Y_\eta|<2^\omega$, so $F_\eta\setminus(X_\eta\cup Y_\eta)$ is infinite, and we can choose distinct $x_\eta,y_\eta\in F_\eta\setminus(X_\eta\cup Y_\eta)$ to continue the construction.

Now let $X=\bigcup_{\xi<2^\omega}X_\xi$ and $Y=\bigcup_{\xi<2^\omega}Y_\xi$; by construction $X$ and $Y$ are disjoint sets meeting each uncountable closed subset of $\Bbb R$, so both are Bernstein sets.

Added: In my experiences they are most useful as a tool for constructing (counter)examples. This post in Dan Ma’s Topology Blog is a good example of such use.

Solution 2:

Notice that the usual construction of the Bernstein set can also yield $\mathfrak c$ disjoint Bernstein sets in an arbitrary uncountable Polish space by working diagonally, similarly to how you construct a bijection between $\mathbf N$ and $\mathbf N^2$: you take one point from the first closed set into the first Bernstein set, two points from what remains of the second closed set to first two Bernstein sets etc, resulting in a sequence of distjoint sets, each of which intersects all but possibly some $\kappa<\mathfrak c$ uncountable closed sets. Then you notice that every (uncountable) closed set contains $\mathfrak c$ other (uncountable) closed subsets, so each of those actually intersects ALL uncountable closed sets.

In many ways, Bernstein sets are "as pathological as possible" for a subset of a Polish space, and as such are often good source of examples of pathological behavior.

Bernstein sets are not only non-measurable (and extremely so: the outer measure of a Bernstein set with respect to any continuous Borel measure is full, while the inner is zero!), but also do not have Baire property.

They are also quite interesting as topological measure spaces. Every compact subset of the Bernstein set is countable, so it's impossible to define a (nontrivial) continuous Radon measure on Bernstein set, which makes them sort of the opposite of Polish spaces -- in a Polish space, any finite Borel measure is Radon.

Solution 3:

This is exactly the Proposition 3.1 in Handbook of Game Theory, Volume 1, Chapter 3, Jan Mycielski.

There is a payoff set such that the Gale-Stewart game is not determined.

Proof: Let the payoff set be the Bernstein set. Given an arbitary strategy, the set of all possible outcomes generated by it is a perfect set(see Brian M. Scott's answer), so for any strategy of one player, there is pair of opponent's strategies that generate two outcomes that are in the payoff set and its complement respectively.

Solution 4:

Since a Bernstein set is nonmeasurable, you can't do it without some form of the Axiom of Choice: there's no "explicit" construction.

An example of an application: show that every measurable set of positive measure contains a nonmeasurable set.