The celebrated Josefson–Nissenzweig theorem—in a special case of a Banach space \(C(K)\) of continuous real-valued functions on an infinite compact Hausdorff space \(K\)—asserts that there exists a sequence of Radon measures \((\mu_n)\) on \(K\) such that the total-variation of each \(\mu_n\) is \(1\) and for every continuous function \(f\in C(K)\) the sequence of the integrals \(\int_Kfd\mu_n\) converges to \(0\). All the recent natural proofs of the theorem start more or less as follows: "Assume there is not such a sequence \((\mu_n)\) but with an additional property that each \(\mu_n\) is a finite linear combination of one-point measures (Dirac's deltas). Then, ..." Although the proofs are correct, it appears that it is not clear at all when this assumption is satisfied. During my talk I will show when (and when not) it is the case that a compact space \(K\) admits a such a sequence of measures. As examples Efimov spaces, products of compact spaces, Stone spaces of some funny Boolean algebras will appear.

This is a joint work with Lyubomyr Zdomskyy.

A video recording of this talk is available on YouTube.